Stability Conference 1986 Proceedings Vol I

be  determined .7 -

m

 Angular motions and   moments

Baji+fc-ior)
toq+öw-fcOpr--iaO^-p'H mzo(u+wq-vr) -mxc
(2)

W+(to-Iw)pq-Mp>qf)+mxG(v+ur-wp) =N+mgxoco80sln^ where m Is mass of the ship,  g 1s the gravitational acceleration, u,v and  w are linear velocity components along the x,y and  z axes, p,q and  r are angular velocity components about the x,y and  z axes, xfi and z« are  positions of the mass centre 1h the body co-ordinate system, X,Y and Z are the components of thé external fluid  and wave forces acting on the ship in the body axes direction  x,y and  i-  respectively, ',„•'„„•'„ an d  '„,a re  the  moments of Inertia and the product of Intertla 1n the usual definitions, K,M and N   represent the  moments about the   orgln o of the  external forces acting on the ship and X,6 and  * are the Eulerlan angles such that the ship 1s given yaw x  about oz, trim 9  about  oy and   heel 4>  about ox. In that order.' The forces X,Y,Z and   moments K,M,N In equations (1) and (2) result from   the Interaction between the ship and the sea and, 1n general, depend  on the time 'history of the motion of the  ship 1n the sea. The  motion  may' be characterized  by the  position, velocity and   acceleration of the ship. In general, as functions of motion variables, the force and  moment are  written Iri the form

X=T(1-t)-R+XKD(AJv1  and D.F including WJD.F)

X|.ç,Y„ç and Z.,- are hydrostatic components  of the force proportional to position of the ship, X^ Y, .. and  Zrj, are  components of the Froude-Krllov force  based  on the Froude-Krllov hypothesis and   also the  moments K.M and N are  specified.by the same suffix as forces. Solution of the hydrodynamlc problems have heretofore been obtained only under assumption of small motion amplitudes. This assumption cannot be used  1n the present case. Instead, as noted  previously, we shall focus oh an exact computation only of the hydrostatic part of the fluid force and the wave displacement force of the wave excitation.  Neglecting all the  terms of; the velocity and acceleration of the  ship and the sea 1n equations. (1) and   (2), the  equations for  Instantaneous equilibrium position of the ship In a  wave are obtained  as follows: (YHS+YFJ<)Slnrfi+(ZHS+ ZFjOCOSaS

+mgcos0=O (MKs+MFj<)co3^-(Nas+NFj<)3lnrf)  -mg(XGCOs0+ZGSinflcosdi)=O (KHS+KFj<)-mgz6CO803inrf)=O

3. ANALYSIS OF FORCES AND MOMENTS For analyzing the forces and moments acting on the submerged surface of the ship In a.quartering sea, we assume a regular sinusoidal wave travelling with amplitude a, wave number k and   phase velocity c 1n the direction of the o€  axis. At the any time t the elevation çw 1s

fw=acosk(£-ct)

+XHs(fo.M)+XFj<(a^.X  îo.X) Y=YHD(AJV1 and D.F  including  WDP)

+Yas(£b.e,9S)+YFj<(a.AXfo.t)  

(3)

 p=pg{-pgacosk(£-ct)

+Zn8(&>,M)+ZFJ<«'^Ur,fo,t) and K=KHD(AJV1 and DP  including WJDP)

+Kas(fo.fl^)+KFj((a^^. i o.t)

+Kas(&.e.^)+KFx(a t Äxlo,t)

(6)

and   the pressure Is approximately

Z=ZHD(AJV1 and DP  including  WDP)

M=MRD(AJV1 and DP  including  W.DP)

(5)

(4)

N=NHD(AJV1 and D.F including  W.DP)

where P g Is the specified weight of the water. In equation (7), the so-called Smith's- effect due to the orbital motion of water particles 1s neglected. It 1s actually possible to take Into account this effect, but It Is.known that the error arising from the neglectlon Is not so great. And   also, the wave elevation C and   pressure p may be  refered  to the  body axes, oxyz by  replacing € and Ç by the components

+NHs(fo.e.$)+NFx
É-Éo^xçosAr-tycosÉ-zsInWsln*

where T' 1s the  thrust of propller, t Is thrust reduction,  R 1s ship resistance, xnD' Y HD an d  Z HD a re hydrodynamlc components of the force proportional to velocity and   acceleration of the ship and the wave.

(7)

--

{ - f o "  x0+ysln#+zcos$ then

- 8 -•'

(8)

£w=ocoskl£o +xcos*-(ycos*-zsin*)sinAf-ctl(9) and

P=pg(Co-x0+ysin#+zcos0)  

(10)

-/Bgacosk[^0+xcosAr-(ycos*-izsin*)sln^-ctJ where £„   arid C Q are the Initial position of the origin of the body axes measured along the ot and oÇ axes  respectively and equation (9) Is approximately Induced under an assumption that the trim angle 0 Is small and    the breadth and depth of the ship are small compared with the length of 1t. The force and moment of equation (6) are given by Integrals  of the pressure gradient over the submerged volume V of the ship. Thé components of the force and moment 1n the  body axes oxyz are

and

(YH3+YFK)Slh^+(ZHS+ZFK)C0S$

H(x)=pgcos#^ ~b8(z)dz

=-njv[(^)sin#+(f)cos*]dV=-pgjnvdV

(MHS+Ntoc)cos^(iN««+r*=K)3ln^

(ID

=njx[(^)sln#+<£)cos*]dV

-JJJv(ysln*+zcos0)^dV~P9lJIv

Flg. 2 Sea surface at port and starboard sides

 A(x)=£ b(z)dz+£ b(z)dz +4-(Zp-z8)[b(z p)-b(z8)]

-^9C0S*i(Zp-2a)Cb8(Zp)-b(Zp)b(Z8)+b,,(Z8)J (16) .0 • -pgsin#[J_ zb(z)dz+L zb(z)dz) P

7* :

-pgsin#l(2p-28)[(2Zp-rz8)b(Zp)-(Zp+2z8)b(z8)]

xdV

the force and  moment represented by

(Kas+KPx) =-JJIv(y($)-z(^))dV —pgJJJ (ycos*-zsin$)dV The Integrals are taken over all volume up to the . sea surface. Making the Ç of equation (9) equal to the ç of equation (8), the sea  surface 1n the body axes oxyz can be given by

fo-x0+ysln*+zcos*=acosk[fo +xcos* -(yco8#-zsin0)sin*]

of   equation  (11)   are

njv d V î 3 t A M d x

(17)

/>gJJjy(ycos*-zsin#)dV • fLH(x)dx 

08)

and

Then we have, from equation (5)

-pgJ   A(x)dx+mg»0 L (12)

 pgJ LxA(x)dx-mgXg»0

Then the sea  surface at port and starboard  sides, z_ P and z . are

J LH(x)dx+mgz38ln^=0

s

£o-x0-b(zp)8in0+zpcos*=acosk[£o+xcos;r; +(b(Zp)cos#+ZpSin#)sin*]

(13)

and

£o-x0+b(ze)sin#+z8cos*=acosk[Jo+xcosAr -(b(z8)cos*-z8sin*)sinAr]  

(15)

(14)

where b(z ) and b(z ) are the half breadth at z and z as shown In Flg.- 2. Since  the.. sectional area A(x ) and   heeling moment H(x) of the station x are given by

<19>

where L 1s all  over the submerged length of the ship. We can   compute the Instantaneous equilibrium position on the righting moment  of the   ship according to equation (19). 4.  EQUATIONS FOR  COMPUTATION RIGHTING MOMENT

OF   POSITION  AND

At .the beginning of   these computations, as noted previously. It Is  necessary to obtain the sea surface at port and starboard sides, z. and z In p s equations (13) and   (14). A  convenient way for computation of them will be to use the perturbation

•ethod. In this Method, the sea surface elevation Is given, by the sta of Infinitely small amplitude waves as follows: (20)

^j^cbCosk($,+xcos;r--(ycos^zslrtd))sln#-çt]

where Aa n Is equal to a/N. For perturbed variables of  A(.,A6,A$,Az and  Az In equations (13) and (14) caused by Infinitely small amplitude wave Aa, we  obtain  A£ o ~xA0-C*(Zp)Agt-C*
(21)

 A£ o -XA0-^(Zs)Aflt-C«(Z»)Azs =AaC08a(Z») where

C*(2p)=b(Zp)cos0+Zp3in0+àkïb(Zp)8in* -ZpCós0jsinJrslna(Zp) Cx(Zp)=b,(Zp)sln#^cos*-akIb ,(Zp)co3# 

(22)

+8in#]sinJrslna(zp) a(Zp)=kIio+XC08AT+(b(Zp)CO8^+Zp3in#)Sin * ] and C# (z,)»-b(z,)cos#+z 8sifi#+okt-b(z 8)8in0 -ZoCôs^JsinArsinatZo)

 AiJ   Ai(X)dx-A^A2(x)dx-AçSj[A3(x)dx L =Aa[_A4(x)dx  A^J LxAi(x)dx-AflJ LxA2(X)dX-A#£xAs(X)dX

=AaJ LxA*(x)dx

(26)

 A{J LHi(x)dx-AflJLH2(x)dx-Ad;(J[rb(x)dx -mgZgCos«l)=A of the final position. The ship hull 1s approximated   by a number of  polygons representing the station of the ship. Each  polygon is In a  plane defined   by a constant value of x 1n the body axes.  Next, we shall think of the righting moment of the ship In a quateMng sea. Although the righting  moment of a ship 1n still water Is, In general, calculated  for thé anlge of  Inclination measured from  the upright condition, as mentioned above, the  position of a  ship in a  quatering sea may be deviated from  the upright, condition. That Is to,,  In the condition with „say, the ship may be balanced  an Initial heeling angle where the righting moment 1s equal to zero as  shown 1n  F1g. 3.  This Is a

(23)

,

C,(z i )=-b'(z 8 )8in#-cos-ok[-b (z 8 )cos* +8ln0j3inArslna(z8) «(2 t ) e»k^0+xcö8Jr+(-b(z 8 )co8#+z8sin*)slnij

Then, for small changes 1n A(x) and    H(x) of  AH(x), caused  by the Az equation (19), AA(x) and  and Az( of equation (21), we obtain

J  AA(x)dx«0 L £xAA(X)dX=0 

(24)

^AHMdx+mflZgA^cqs^O and

 AA(x)=Ai(X)A{0-A2(X)AS-A3(x)A^-A4(x)Aa  AH(x)*Hi(x)A^Ha(X)Ae-Hs(X)A^-H4(x)Aa

Fig. 3  Righting arm In a wave different condition condition condition Then given by*

(25)

where  A-(x), ... , H^(x) are the  coefficients of  perturbed variables AÇg,A9,A$ and Aa which are given 1n analytical form  by making perturbations of equations (13), (14), (15) and  (16). The .terns of them, however. Is too  long to describe here. Finally, equation (20) can be decomposed as

point with respect to the starting to calculate 1t. This initial heeled : relatively correspond  to the upright of the ship In-still water. the righting arm GZ(4) 1s geometrically from Fig. 3,

GZ(#)=BR(A#)-BG8in(A^ 

(27)

where BR and BO are the  same as one given by the traditional definition and   have to satisfy the following «onditlen

, 10 -

So that, we can obtain a formula to compute the

righting moment mgßZ($) as  follows:

+0dx+m0%8ln(/+$) 

(29)

'where

2£=OB-BG

(30)

On the other hand, the  slnkage C Q and   trln angle 6  caused  by the angle of Inclination of *. Fron equation (26), the equation Is given by

 A^Ai(X)dx-Ae£Aa(X)dx=Atf>JLA8(x)dx  A4LxA»(X)dx-Aé^xAa(x)dx»Arf;J|_xA3(x)clx

(31)

The maximum righting anas  for   various ratios  A/L against heading angle X •»"• shown In Fig. 6, those for various ratios  H/A are In Fig.7, those  for various ratios A/L In constant H/A are In Fig. 8 and those for various B6 are In Fig. 9 where the maximum righting arm In wave with cr est anldshlp Is divided  by that of B6 equal to 3m In still water. Finally, to measure the heeling moment acting on the heeled model towed  In quartering seas, a captive model test  1s carried  out for the  trawler's  model In the condition wher e the model has a given heading angle equal to drift angle and 1s free with respect to   slnkage and   trim. Fig. 10   shows an example of the experimental results compared with calculations.

still water Here, 6 should  be measured from the Initial heeling angle o>.. " ' ' 5. RESULTS OF NUMERICAL COMPUTATION In obtaining solutions, the ship  Is now assumed not to deviate from   a given heading angle x and to  be In the relative position defined  by €Q which Is the position of ship to wave at the time equal to zero. The slnkage CQ , trim angle 8 and    heel angle which are caused  by  the wave excitation, are given as a  solution of equation (26). Then, we  GZ, the slnkage Ç. . can compute the righting arm  0 and trim angle 0 which are caused . by thé angle of Inclination . Some numerical computations are presented here for two ships: One 1s a container ship and another Is a trawler. First, for the pure loss of Intact stability of the ships with several heading angles,  the righting arm curves, slnkage and    trim angle are given 1n Figs. 4 and 5  compared with that in still water where the wave to ship length ratio A/L is 1.0 and the wave-length ratio H/A 1s 1/20. The Influence of heading angles on the righting arm   curves can be significant. That 1s to say, thé righting arm    curve In a   beam  sea: with crest amldshslp 1s about the same as that in still water. But  they.are smaller for smaller heading angles arid the smallerst one 1s  that In a following sea. The variation of the  righting  arm is more remarkable for the container ship than that for the trawler. The  nature of GZ in a  quartering sea seems to be strongly dependent on the ratios B/D and B/d of huV>  form. Next, ;. let us focus on the maximum righting arm of the container ship with.an arbitrary heading angle X about the Influence of wave to  ship length ratio  A/L, wave-length ratio H/A and the change of B G . '

'•

•'•"

:.-.-.'

V 

:

••''

' - .• : ' ,

' ' ' : • • . .•':,••• •'.

•'•"

• •' •

HEMS OF CONUI Ngfl Ip p (ml B (») Im ) 0 d In» ) gross tormags

SHIP I IVO 19.0 M «.« 4.0 90

6 CD

s trim angle

K

 — ^ — : slnkage »till woter _

S water/

Fig. 4  Righting arm    curves, slnkage  and  trim In wave with crest amldshlp for a container ship »till water

iifHt or norm BAP  HOUO i„ •

o

Is ) i t « LOK l a l , SJO U N

I») lai

4g •=

I -a

».«o oji» g S ta am s % •to .
i trim angt*

 ——: i slnkage

Fig. 5  Righting arm   curves, slnkage  and  trim 1n wave with crest amldsMp for a trawler

10*. . x

60*

'

90*

Flg. 6 Maximum righting arms for various ratio A/L

Fig. 7 Maximum righting arms for various ratio H/L

6. CONCLUDING REMARKS By reference to the diagrams of Figs. 4 tó 10, the following deductions can be made: (1)  All thé maximum righting arms 1n wave with crest «midship are close to  that 1n still water as the heading angle of à ship to wave approaches to a  beam sea. (2) For the maximum righting arm, the rate of the change 1s  small for the  range of  heading angle 0°
7,  ACKNOWLEDGEMENTS

30' 

x

*>'•

Fig. 8. Maximum righting arms for various ratio A/L In constant H/A

1

REFERENCES

A/l.  I.HM.I/20

z r/ /

*^fS

r  .,

s-  /*" <

S* h»adlng onglt

Fig. 9 Maximum righting arms for various

4

 by the Shipbuilding This research was supported  Research Association of Japan (Research Panel  RR24). The author would like to  thank the RR 24 panel  members for their Informative discussions.  The author would also like to express his sincere thanks to Dr. K. Hasegawa and all colleagues who cooperated In this study. .

U 0* - 0.« 0.» 1.0 rotative position of ship to wav«. ç,

o

[1] Chou, S.,  Oakley, 0. and   Pauli1ng, R., "Ship  motion and   capsizing In an  astern sea",  U.S. Coast Guard, Office of  Research and    Development, Report  No. CG-D-Ï03-75. : : [2] Grim, 0., "The ship In a following sea", DTMB,  AD, No. 458. .'•.-•..•; [3] Hamamoto,  M; and    Noraoto,  K.,. "Transverse stability of ships 1n a following sea",. Proc. of Second International Conference on  Stability of Ships and    Ocean Vehicles, Tokyo, Oct. 1982. [4] Kerwln, 3. E.,, "Notes on rolling In longitudinal waves", I.S.P.. Vol;2, No.  16.'1955.; [5] Paulllngi'J. R., "The transverse stability of a ship 1n a longitudinal seaway", J.S.R., Vol.4,  1961. [6] Shipbuilding Reaserch'Association, of  Japan, Reports No.91R, 99R and   108R. [7] Welnblum, G. and St. Denis, M., "On the motions of ships at sea", Trans. SNAME. 1950.

Fig. 10 An example of experimental results

- 12 -

H. Hamamótó graduated from the University of Osaka Prefecture, Japan, In.1959. He then worked at the Research and Development Centre of Japan Defence 'Agency as a research engineer for seventeen years. He received the masstar's and doctor's degrees of engineering 1n • naval architecture from Osaka University, Japan, In 1966 and 1975. He was appointed to Osaka University 1n 1976 and has been a  professor at that University since 1984. He Is Interested- In ship motions 1n following seas and hydrodynamics for manoeuvrability of ships.

••  .-"13'.•

Third International Conference on Stability of Ships and Ocean Vehicles,  Gdarisky Sept. f986 Paper 1.3

ON THE PROBLEM OF. PEAK-ROLL-RESPONSE OF A SHIP UNDER A   WIND-GUST L.R. Raheja

ABSTRACT A modification over the conventional energy balance method for the deter mination of peak-roll-response under a wind gust is suggested so as to take some of the characteristics of the seaway directly into account, it is pointed out that the kinetic energy of the ship rolling in a seaway, just before the windward heeling starts is an important factor in the determination of the maximum angel of roll.  The conventional assumption that the wave action ceases to exit when the gust strikes, is modified. It is now assumed that the wave action continues even after the gust strikes and ends just before the windward heeling starts. Consequently, the kinetic energy as mentioned above could be estimated from the seakeeping analysis of the ship in place of conventional righting moment curve and weather criterion. Finally, a graphical method to determine the maximum angle of roll using energy curves in place of conven tional moment curves is proposed. 1. INTRODUCTION One of the essential requirement of stability deals with the maximum angle of roll suffered by the ship when a wind gust strikes the ship. This angle is estimated conventionally by an energy balance approach using the curve of statical stability and a weather criterion which may be different in different nations  [1,2]. Besides the differ ences in the assessment of wind gust moment, various national criteria also differ in their approach in the calculation of the in itial angle of roll at which the gust is assumed to start acting. The problem of rolling subject to the influence of a sudden wind gust is quite complicated one . The ship is supposed to be rolling in waves before the gust strikes. The gust moment itself varies nonlinearily with angle of roll.  Consequently, the for mulation of the problem requires an unsteady, dynamic and nonlinear mathematical model [3] and therefore complicated. The conventional energy balance approach seems inadequate as it takes no direct consideration of the dy namics of the ship rolling in irregular waves before it is influenced by the gust. However, this approach still remeins a con venient alternative for the calculation of the maximum angle of roll under a wind gust. In the present work, an attempt is made - 15 -

to modify this conventional approach in order to incorporate some dynamic aspects of the pre-gust motion through the assess ment of the kinetic energy of the ship be fore it starts heeling windward. The modi fication is directed towards making the calculation more realistic and reduce the dependence upon arbitrary weather criteria. The conventional approach is discussed from the point of view of kinetic energy of the ship. The modification in the assess ment of the same is explained. Finally, the calculation of the maximum angle of roll using the modification is described'. The proposed method of calculation makes use of energy curves in place of conventional moment curves as the former are simpler and convenient for the purpose. 2.

CONVENTIONAL ENERGY BALANCE APPROACH

1 represents the usual diagram for the conventional energy balance method to calculate the maximum angle of roll under a wind gust . In this approach, the ship is assumed to be rolling in waves and is heeled to an angle f. when the wind gust strikes and simultaneously the action of the waves ceases to exist. The wind moment W is as sumed to be constant and is represented by the straight line CF. The wind moment as well as the angle f, are prescribed by a weather criterion. Fig.

i

A.QZ .^gM

c W D K,Ä| F

f

IVB W,tâ0 .

vp

r

Si

Pig. 1 Conventional Energy Balance Diagram The maximum angle of roll y>_ is then calcu lated by the equation

Tm

J    [gif) - w } &f  - o

-h

(ï)

where  gif) ie restoring moment of the ship. In terms of areas, the above expression is finally equivalent to  AABCK » AKEF The equation (1) can also be written as 0

} (g(jp) - w } dj» -

- j {  g(y>) - W ) &y> 

(2)

0 The left side of this equation represents the AABCDOA which in turn represents the total energy of the ship in upright position just before it starts heeling windward and should be considered a major factor in de ciding the maximum roll response f> .  This Kinetic energy is gained by the ship partly  by the action of wind gust oh the ship i.e.  AOBCD(-K, say) and partly by its motion in waves i.e. AAOB(»K2  say). It is the esti  mation of latter in which the modification is proposed. 3. PROPOSED MODIFICATION

quency. This is a direct consequence of the assumption that the wave action ceases to exist at position B i.e. when the gust strikes. The estimation of K_ in this way is perhaps oversimplified as it depends upon pre-gust  motion which is its origin, only through the value of f. prescribed by the weather cri terion. The pre-gust motion of the ship must  be that of rolling in an irregular seaway and its characteristics must be known from the seakeeping analysis of the ship. In order to use this information, we modify our basic assumption. It may now be assumed that the wave action continues even after the gust strikes and ceases to exist just before the windward heeling starts, i.e. at the mean  position 0 (Fig. 1). Subsequently, the kin etic energy K, gained by the ship due to its  motion will now be given by the seakeeping analysis of the ship. The estimate of K, could now be modified in one of the follow ing ways in the given sea state,

(i)

2 x "l/3 where

" I

where JP . is the significant roll amplitude and uy may be taken as average frequency of the séa spectrum or the one with highest energy density ordinate. The modification of the assumption and therby the'estimate of kinetic energy K, should make the calculation of V    more 'm •£ realistic. The dependence upon the weather criterion is now reduced and f, in the weather criterion may accordingly be modi fied. 4. MODIFIED CALCULATION PROCEDURE One requires to calculate the value of  f, and W from weather criterion, duly modi fied, as because the value of f. will no longer be used to calculate K». This gives ;

In the conventional diagram (Fig. 1), K, appears to be equivalent to the maximum kinetic energy of the ship executing natural roll osscilatiohs with an amplitude JP. in calm water provided g(J°) is considered as usual linear in they, range, and can be written as

l   2 K, n "2 ~ T*»- --?i where I is virtual mass moment of inertia of the ship in rolling and u n its natural fre

- 16

 <*V J"l/3

K

i

-•

W

J*L

 Next, one calculates K, from seakeeping ana lysis of the ship as described in the pre vious section. Let us take 1 rj It would now be convenient to use integral curves i.e. the curves of energy instead of curves of.moment as used in the conventional diagram, in order to find the graphical so lution of the equation (2) for f  m . Let us define .

- f  gif)

vr>

[3] Odabasi, A.Y.i 'Roll response of a ' ship under the action of a sudden ex citation'. International Shipbuilding Progress, Vol. 29, p; 327, 1982.

df

f2(Jf) « J    Wdj» Equation (2) can now be written as K K f 

- i - 2 =- i<*v +

w

Rearranging the above equation, we can fi nally write f  ( + f 

l JV - W

where f 3

Professor, Department of Naval Architecture Indian Institute of Technology, Kharagpur, India Presently at Institut für Schiffbau der Universität Hamburg, Hamburg, West Germany

3 '

•* Kj + K2.

,

 /^r f 

lli——71

'.. ^<^

•

t,

^

Fig. 2 Calculation of W>_ with modified •

•,

» m

•

approach using energy curves  Now, the maximum angle óf roll V  is ob'm tained graphically as shown in Fig. 2, which is self explainatory. 5. CONCLUSIONS The modified approach suggested above - makes the basic assumption more realistic, - takes into account the sèakeeping characteristics directly, - reduces the dependence upon weather criteria which suffer of arbitraryness and therefore is expected to make the cal culation for peak-roll- response under a wind gust more rational and realistic. 6. REFERENCES [II

[2]

'Intact stability, including analysis . of intact stability casuality records weather criterion', IMCO Document STAB 27/5/4, Submitted by Japan in 1982. 'Intact stability, including analysis of intact stability casuality records . weather criterion', IMCO Dosument STAB 27/5/3;(submitted by USSR, 1982.

--•17 -

Third International Conference on Stability of Ships and Ocean Vehicles, Gdarisk, Sept. 1986 STABILITY AND PARAMETRIC ROLL OF SHIPS IN  WAVES I.K. Boroday, V.A. Morenschildt

1. ABSTRACT In this report, the results of an experimental Investigation of the condi tions giving rise to the ship parametric roll in regular and irregular waves are  presented. It is shown that the main cause of this dangerous phenomenon is pe riodic variation of metacenter position due to the ship heave in relation to the wave which; as a rule, takes place if the ship has a large broadside flare in way of the water-line. An approximate method is proposed for computing the parametric roll of the ship on the beam wave. The data obtained by this method are in satis factory agreement with the experimental results. This method increases the reli ability of predicting the ship Tolling  motions in steep short-period waves. 2. RESULTS OP MATHEMATICAL SIMULATION OP THE PHENOMENON It is well known that ship stability at large heeling angles is the most impor tant characteristic of her safety in still water and in waves. In particular, subs tantial broadside flare makes for an in crease in stability. However, as is shown  below, this kind of flaring can lead to such an extremely undesirable phenomenon as.parametric rolling motions. Though a relatively large number of works is concerned with the investigation of parametric roll, up to now there has  been a diversity of opinions about the causes of this phenomenon and possibility of its occurrence in irregular waves.  Various researchers stated that the  possibility of occurrence and the charac ter of parametric rolling depended- on the following factors: involvement of the ship in orbital wave motion, periodical changes in stability due to relative heave, center of gravity elevation above water-plane, form of stability curve, and  magnitude of linear or quadratic damping. Identification of the effect of each of these factors by model testing in towing

- 19

tanks requires too muoh research effort. Therefore, in order to evaluate the ef fect of individual factors and to take the most important of them for subsequent  physical investigation, it was decided to simulate on an analog computer a set of equations giving the most general repre sentation of parametric rolling. The re sult of this part of work which are given in £1] have shown that'. - parametric roll whose frequency is half the frequency of excitation arises in the cases when the ship heave in relation to the wave leads to changes in the position of transverse metacenter; - the amplitudes of parametric roll do not increase infinitely; their intensity depends on the linear and quadratic com  ponents of damping; - parametric roll can exist both under regular and irregular excitation (see Figure 1); - in the frequency range typical of the conditions which give rise to parametric . roll there is no immediate effect of the wave-induced excitation leading to oscil lations at the excitation frequency, therefore, the rolling motions within this frequency range can be described by the following equation with sufficient accuracy for practical use: 2 

Q+(m+K6 o )6     + n  e (l-6sin(àt)d=0 

d)

3. RESULTS OP MODEL TESTS IN REGULAR  WAVES In order to verify the results of the mathematical simulation, two models were tested oh the beam wave at- zero speed in the towing tank of the Krylov Shipbuilding Research Institute. The form of model 1 was close to the side-walled one, whereas model 2 had a large flare in way of the water-line. Figure 2 shows the  positions of transverse metacenter, Z m » as functions of the draft, T , for the

two models. The loading of the both mo del e was performed  ao that the natural heaving period, T«- , was equal to 1a and the natural rolling period, T 0 , wag equal to 2a. In the wave range, J. , ge nerated  by the wavemaker (1,5  aéJLé  10m} suoh values of the periods made it pos sible to investigate both the main resonanoe (that at exoitation frequency) and  parametrio resonance conditions. Prior to eaoh test series, the dependence of the nondlmensional linearised roll damping coefficient, j ^ , on the roll amplitude, Q0 , was determined using the method of free oscillations. The curves obtained for the two models were close to each 'other. ;',"''.'''  As is seen from figure 2, the vari ation of the draft from the designed  one for model 1 was not aooompanied  by a ohange in metaoenter position, whereas for model 2 it caused  a significant ohaqge* Therefore, based  on the results of the  mathematical simulâtion, one would expect the ooourrenoe of parametric rolling for  model 2 and its absence for model 1. This was fully borne out by the:model test re sults, see Figure 3. It oan be seen that  model 1 had no parametrio roll, whereas for model 2 the roll amplitude was as large as ~ 25°. 4. 1ŒTH0D/FOR COMPUTING PARAMETRIC ROLL  AMPLITUDES  According to [ 2] ,  the condition ne cessary for the occurrence of regular pa rametrio oscillations is provided  by satis fying the inequality«

•: ;'.(>,•$£.•,;

:

«>

In this oase, parametrio roll would ooour In a frequency range desoribed  by the formula:. '.

'is,the ship stability during a heave re- ,  mains essentially poeitive. For the real : values of n ß the frequency range where  parametric oscillations oocur is found to  be very narrow. Por example, for &  h»0,2h it satisfies the condition 0,95 £

Inequality (4) is valid for : £*< i, that

C  1,05 .

(5)

The parametric roll amplitudes, t'o pa r . in the frequency range satisfying the oondition (4) can be calculated using the formula  (*2l»

where

 X~

CO

2% '

The maximum amplitude of parametric 1 (parametric résonants) roll for  2nd is:

P«t V (A,   'mai

&Çû

0

HK 

m » tod.  (7) <

It should  be borne in mind that pa rametric oscillations have the frequency Ctf/2, hence, the amplitude of angular velo city la

i'"=Oi).

(8)

So,  for the prediction of parametria roll knowledge of the values of 0 , m, K is required. If the amplitude of the ship heave in relation to the wave, Q,0 , and the rela tionship between metaoenter elevation and the draft ïn, = -f (T) are known then the value of B is defined  by the expres sion:

S~

Q*l4<\l  -iffl'aV^m^jfy^ »XT (3) Since the condition fll ^ R Q is always satisfied»damping has practically no effect on the boundaries of parametric oscillation ooourrenoe, and condition (3)  pan be represented  in a simple formt

&

 dim

h 

: •*• •'• '

(9)

ia where dV   * he a1?«11181, coefficient of a straight line approximating the curve %ti,=-4(T) tor  the region of the stillwater wat erline with respect to the axis of drafts..•'•'..• It should  be noted that the possibi lity of Ooourrenoe and intensity of para  metric oscillations does not depend on whe ther ship stability increases or deoreases with an increase in draft. Equation (1) suggests that the plot

- 20 -

of the damping moment as a function of roll amplitude is a straight line which does not pass through the origin, and the na a a  plot of the coefficient  %^--f(9t) similar form. The coefficients m and « then oan be expressed in terms of -yL by the formulae:

^ ^ ' S  r-ljnfr«»,

frequency CO" 2/1* , that is,formally, Sg(.2r\0 ) » 0. Yet, parametric oscilla tions occurred if the harmonics within the range determined  by the conditions (4) or (5) had enough energy. Therefore, the criterion  (11) should  be extended by writing it in the form:

  O0)

where  (r^j)* is the value of the . nondimensional roll damping coefficient with öo- 0; 4 ^ is the angular coef ficient of a straight line approximating the relationship -y^» =:J!fQ e )  with res pect to the v0  -axis (whenn calculating the value of % the derivative   dJ0 QW*  be expressed   In radians). should   j(0,)  for The relatior»hip "typ —  real ships may be rather far from being a straight line. However, parametric rolling is only dependent on the value of K   which corresponds to a given amplitude, and the  at other points is practically curve form  of no importance. Therefore, for the cal culation of parametric roll amplitudes a  method  of successive approximations can be used similar to that used   in calculations of usual resonance in case of nonlinear damping  [})•. . Together with the experimental values of roll amplitudes» Figure 3 gives their  predicted values; Outside the frequency range governed by the condition  (5) the calculation was performed using the con ventional method f3j, whereas within this range use was made of the method described above. The agreement between the predicted and experimental values is quite satisfac tory.  As a criterion of the possibility of  parametrio roll occurrence for a ship cha racterized   by linear damping in the follo wing or head seas, I.K.Boroday has proposed the inequality

^V-o ^y^KO» 

(12)

where  Si(npQt") is the maximum va lue of the spectral density of S in the frequency range determined by the condi tions (4) or (5). Based  on the above considerations a  method for predicting ship parametric roll in irregular waves was suggested. The con cept of an equivalent wave was introduced as the basic one which denotes a regular wave with the frequency CO = 2 n e and the amplitude %e  determined by the energy of an 'irregular sea. spectrum in the range of parametric resonance, namely,

»v/zalr

(13)

6)=lJS(Zn e ) 

where 7\ —

J) f J

Ss(<4dQxO,2n e Ss  (  nX"

4)

= mean value of the sea & s(n m*) spectrum in the frequency range defined  by the condition ( 5). For this equivalent wave, the ampli tude of ship heave in relation to the wave and the relative change in stability, 0 , are computed Aising formula  (9), and the. possibility of the occurrence of para  metric oscillations according to inequa lity (11) is verified. Further computa tions of parametric rolling are performed only if this inequality is satisfied. It is advisable to use the grapho-analytical  method for these computations which are  performed in the following order. Several values of'. the parametric roll amplitude Ö are prescribed, for each 0

-

. •

it

is cal culated   by formula (10), and the relation-? of the values the coefficient . (C

where Sg(2"8) is the spectral den sity of . 6 at the frequency  Ci)~-'2n$ f This criterion had to be additionally improved, since in the mathematical simula tion, when excitation spectral density was specified as the sum of a finite number of harmonics, parametric roll was also obser ved when there were no harmonica with the • •••' ;-:',

.',;•'.'"''-•"•

',.'.. :. :--:-;:

- 2 1

ship .*-,.ƒ f ^ with

X&

 É  Z

f rod')

(15)

is plotted. (1 For the value of , 0  corresponding to the. "equivalent wave" and for several

values of K  , the amplitude '0 pa,t as a o funotlon of the frequency CO is calcula ted using equation (6). The results of the calculations are represented  as the square of the transfer function modulus of para  metrio roll with respect to the value of

(sri M .= V*«

(16)

For several values of K the parametric roll spectral density S«  variance and  >— pax *  Me are calculated '9  c pat

:lVl^^.^l^WV

 ai*tn\/l*$t

2);=]  .:$;(»)**>., where

The spectral density of the total roll angular velocities, $•  -J-((û) , is calculated  as the sum of two components, one of which corresponds to osoillations at exoitation frequenoy and the other re fers to parametric roll. Talcing into con sideration the characteristic properties of each type of oscillations, one can write«

(17)

Ls the square of "" the transfer func tion modulus of the Value of §  with res  peot to the ship heave in relation to the wave; a is the square of the transfer function modulus of the heave with res  pect to the wave elevation. On the same diagram where the rela tionship , lf«j ( Dou sed  in the computa^ tions has been plotted, and at the same scale, the relationship 'IL  =»-ƒ f K ) obtained  by equation (18) is plotted.The intersection point of the two curves de fines the final value of K  , for which, using the above method, the resulting spect ral density and   variance of the parametric roll are calculated. For several values of the relative damping coefficient, *}À  » Y O ^ ' A * • * n e spectral density of the ordinary roll ha ving the exoitation frequency, SQ , is calculated. The calculation is oarried   out aocording to [3],  with the difference that the frequenoy range determined by the con dition (4) is excluded. For the chosen,values of y ^ the spectral densities of the total roll , S$ , are plotted   as functions of the excitation frequenoy 6J . Outside the fre quency range defined by the condition (4) they are the spectra of the ordinary roll, and within that range they represent para  metric roll speotra. The spectra Sa have the form shown in Figure 4. The area  bounded by the curve and the axis of CO define the variance of the roll angles, 2>0 , for a given value of 9$ - 22 -

The variance of thé total rolling velo city, fig , is defined  by the area  bounded  by the ourve Sx (&>) and the CO -axis. Further determination of the energy-statistioal linearized coefficient of damping, significant amplitudes, and   mean rolling period   is carried  out in the same way as for the ordinary roll £ 3j. It should   be noted that the spect rum shown in Figure 4 is essentially a  pseudospectrum, sinoe it represents the distribution of the energy of oscilla tions by excitation frequenoy, and hot  by the frequency of osoillations, as the spectral density definition implies. These pseudospectra can be used   to calcu late the varianoe of oscillations, but not to make comparisons with the spectra obtained from mathematical or  physioal simulation. Por the parametric roll spec tral density to be applicable for the  purposes of comparison, this must be shown at the actual parametric roll fre quency, that is,at half the frequenoy de fined  by the oondition (4 ). Since the frequenoy range of oscillations is half as wide as the frequenoy range of exoita tion, the ordinates of parametrio roll spectrum should   be doubled   in order to retain the varianoe. The total roll speo trum, S_. , takes the form shown in Figure 5. The angular velooity spectrum is then defined  by the expression:

*<(«>)-. #-«>'

(20)

5. RESULTS OF MODEL TESTS IN IRREGULAR  WAVES Since the proposed computational me thod involves a number of assumptions, it was considered reasonable to carry out the tests with model 2 in artificial ir regular waves generated   by a pneumatic wavemaker. The wave spectra were varied . within wide limits, both in frequency

content and Intensity. The test results hâve shown that whenever the spectrum had enough energy to satisfy the condition (12^ there was a sharp increase in the roll amplitudes, and the roll mean period was close to twice the period of the waves, which indicated to thé occurrence of para  metric oscillations. By way of additionài illustration, one example will here be examined. In the towing tank, the model was tested in waves having the spectrum shown in Figure 6. It oan be seen that the maximum of the energy of A hi8 spectrum is concentrated near the frequency  0*Zliß.,and that near the main resonance ( the roll amplitude of 3 %  pro  bability of exoeedânce ih this case must . not be in excess of 2°. In practice, du ring the model test the roll was very in tensive and the amplitudes exceeded 30°. Figuré 7 shows spectral densities of rolling anglesi thé expérimental spectral density, that calculated following the  proposed procedure, and that calculated without taking into consideration paramet rio oscillations. Aocording to the experi  mental data, the rolling motion variance wàfii ft$ •'* 151 deg. Calculation by the  proposed method gives iÖö» 108 deg.2 which leads to a discrepancy of the order of 20 per oent between the roll amplitudes.  At the same time, the calculation without taking into account the parametric oscilations for the given wave spectrum yields thé variance'Jfy. " 1  dég.2 which is by two orders of magnitude leas than the éxperi\ mental value. o

ric roll. It has been confirmed' that this kind of oscillations can exist both in regular and irregular waves. In order to exclude thé possibility of this phenome non, it is advisable to design the abovewater hull form so that during the ship's heave in relation to thé wave the posi tion of metaoenter remains possibly con stant, and for this purpose the Undue flaring should be avoided. The proposed computational method makes possible con siderable inorease in the accuracy of  predicting ship roll in the waves of re latively short periods. 7. NOMENCLATURE

/7g> natural roll frequency ft

 m

« changé in transverse meta centric height h ( **M) due to the ship heave in rela tion to the wave ,

(c • coefficients for the linear and quadratic components of the damp-, ing moment, respectively

O) » wave frequency  yLa." nondimensional roll damping coeffi cient > 5-' •" spectral density "of ^ j 2)_ • variance of

X,

8. 'REFERENCES 1. Morenschildt, V.A., and Smirnov, B.N., . '•; '  "Analog computers in the investigation of subharmonic rolling motions of a ship in waves", Conference on computer technique and advahoed scientific inst rument at ion in ship hydfodynamio s. Iroceedings, Vol.1, Oct.2-5, 1984. 2. Kerwin, I.E., "Notes on rolling in longitudinal waves". Iht.Ship.Pr., ; Vpl.2, No.16.

3. "Handbook of ship theory", edited by  Voitkunsky Y.I. V.2, "Statics and oscillatory motions of the ship" (in Russian)., L., Sudostroehie, 1985.

6. CONCLUSION Taking into consideration pärametrio oscillations substantially increases the accuracy cf the prediction of ship roll ^ m d è r the action; of comparatively short-period Wavesi The investigations made it  possible io establish relationship between the ship stability characteristics and the  possibility of the occurrence of paramet •- 23 -

I.K.Boroday Born In 1933t I.K.Boroday graduated from the Leningrad Shipbuilding Institute. He specializes in ship seakeeping and dy namics in waves. .,-•''

 V.A.Morenschildt Born in 1924, V.A.Morenschildt gra duated from the Leningrad Shipbuilding Institute. She is engaged in ship seakeeping and ship motions stabilization.

Krylov Shipbuilding Research Institute, Leningrad, USSR

Krylov Shipbuilding Research Institute, Leningrad, USSR

Figure 1. Sample of mathematical simulation of ship rolling motions. a) Regular exoitation; b) irregular excitation; 1) waves; 2) parametric roll; -3) heave in relation to the wave.

- 24

"Tö

fco  f-m',cm

26" 4m,C»

Figure 2. Transverse metacenter position as a function of draft, a) Model 1; b) model 2.

>

10

li 1

0

0 \  1 i- ; Ji

I l l 

.1 T lL l

Figure 3. Roll amplitude $A as a function of the wave frequency   aj • a)'Model 1; b) model 2; • , oscillations at the frequency  (Oj o , oscillations at the frequency 6J/2; '. , predicted.

 be )iadj us

J

/ « V  \ V 

1 ;..•••."•:

"

.

.1.

\ \

1

r—^_ 1

Figure 4. Pseudospectrum of total roll as a function of excitation frequency. , spectrum of ordi nary roll for $Q • const.; — , pséudospect rum 'of parametric roll for IC * const.

- 25 -

O

0J,4 -I Figure 5. Actual apectruro of total rol) aa a function of oscillation frequency.

_

gjo e!ne 

o,y'

Figure 6. Irregular wave spectra. 1) Prescribed; 2)obtalned.

W.-5"

- 26

Figure 7. Roll spectra. 1) Computa tion by the proposed me thod; 2) computation with out taking into account parametric oscillations; 3) experiment.

Third International Conference on Stability of Ships and Ocean Vehicles, Gdaiisk vSept   1986 Paper 1.5

 APPROXIMATE DESIGN PROCEDURE OR NONLINEAR ROLLING IN ROUGH SEAS

 V. Shestopal, Yu. Pashchenko

2.   DESCRIBING OF THE DIAGRAMS.

 ABSTRACT

The two diagrams are constructed    for each sea condition. The first diagram  de termines roll angle variance D g and the second   - roll velocity variance D g for different values of breadth B , frequency of free rolling oscillations CJ  and B  damping coefficient  2^e«  As an example in Fig. 1 and Fig. 2 the diagrams for sea condition force 7 are given. Besides, in those figures the tech nique of de-termination  D e and Dg for ship wlthB=i15-5 m, 0) e  =0.645 s"1 , 2i? e =0.087 s"1 is shown. The results of calculation of the rolling parameters of the given snip at sea condition force 7 are shown in /1 /. These values are represented  in  Table 1 (first line),  while the diagram values are represented in the second line'. Thus it may be  stated that  if the rolling of ship is described by  means of linear differential equation, then the use" of diagrams require no other calculations, except the preparation of  initial data. In a general case the results ob tained must be considered as zero estimate of rolling characteristics, necessary  for the calculations of statistical lineariza tion coefficients of restoring and   damping

This procedure is based on the Joint use of the auxiliary diagrams and   program  med microcomputers. The diagrams enable to determine the rolling characteristics in the linear approximation and    further to Improve them   by the method  of  successive approximation in terms of statistical li  nearization*  Application of programmed microcompu ters provides  for the  approximation of nonlinear functions by  polynomials, as veil as calculation of the coefficients of statistical linearization of restoring and damping moments1. The avantages result in almost comp lete elimination of manual calculations, considerable shortening of calculation ti  me and   securing the  sufficient practical accuracy. 1. IHTRODTOTION The determination of nonlinear  rol ling characteristics in irregular waves is executed now by means of statistical line arization method. It is known that statistical equiva lent gain factors, describing the form  of  both restoring and    damping moments, are the functions of output coordinates of the system. Therefore one should execute  the calculation either by graphoanalltical me thod or succesive approximations'. The new procedure is offered for the reduction of volume and   time of calculati ons of statistical characteristics of m o tions. It is based on using the auxilliary diagrams and programmed microcomputers of different types'.

 moments'. 3. STATISTICAL UKBARIZATIOK'S COEFFICIENTS. If a  diagram  of  statical stabili ty will be approximated by means of a po lynomial with respect to odd-<«numbered deg rees of 0 i

- 27 -

0.900 0.800 0.70Q

150 joo 5 0

0.12  Û.\A   0.16 0,18 0.20 Fig. 1.  Diagram for determining roll angle variance

0.12  0.14 0.16 0.18 02 0 Pig. 2.  Diagram for determining roll velocity variance Table 1.

Bl

' ? * • " •

&

.8".

Ös%

^3%

<3

T?

dag a

deg

deg

deg/s

deg

deg/s

VB

B

109.7

46.1

13.0

8,51

27,6

18.1

0.65

9.66

100.0

48,0

12.5

8.66

26,4

18.3

0;69

9.10

- 28 -

a

In th is cas e, the whole ca lcu lat ion of l i  nea riz ati on coe ffi cie nts are reduced to the inp ut of Dg and I)*, pre viou s appr o ximation in to the memory re gis te r of a microcomputer. After ca lc ul at in g (O3 and 2 v 9 i n the first approximation, we again apply to' the diagrams, using scale CJ e  for input by Wg t ntà curve 2 ^ e fo r inp ut by 2 V*8 . Thus we determine the parameters of nonlinear ro lli ng in the f i r s t approxi mation. Then the procedure i s repeate d un ti l re su lt s are obtained with accuracy required for our purposes. In some cr.ses, es pe ci ally fo r large sea condition fo rc e, the process of suc  ces siv e approximations may be div ergent. For improving the oonvergency one can make use of the recommendation  /3/".  When calcu lating statistical linearization coeffici ent  (JÔsn BJli 2V er% one should take as qu al ity arguments not D e  and D Q  of the previou s approximation, but value s de te r mined by formulas

s

• 0,9-» a a 8 + Q59 then the formula of linearization's fioient (ôg  becomes

coef-

The coefficients Q { can be obtained from   the condition of equality of  values of tLe true curve 1( 6) and   function ap   proximating it in three points over a ran ge of angle of heel from nought to 6 max • The corresponding system  of  three equa tions with three unknowns is easy solved with the help of programmed microcomputers of different types. The program, instruction and    control example are shown in /4/ . From   the square-law linearization of resistance

W(é.y-wê*Sijuô

ö)--

we have the formula for coefficient

When pl ot ti ng the diagram the f o l  lowing assumptions were used. T. 'Ship is turned broadside on to the twodimensional irregular seas which speotrum i s determined by formula of 12 ITTO

If the  damping coefficient values, obtained experimentally in form 2v*e m f(&m) are used for the calculation of  rolling, then equivalent statistical linearization coefficient is determined,by the same f function, but the  argument being equal

;.i.9Dif "à' ' z'-

• :SsW-s.ioMo'V 6V ( - ÏÎ^V) (5)

I n talis ca se , f or the program f or co  ef fic ien t 2 9 Q composition, ƒ function must be approximated with a polynomial for degrees 8 , as stated above.

where H y^ - significant wave height (cor responding to 14% probability of  excess),

4 . DESIGN PBOOEPURB PARAMETERS 0 ?

E0LLIH5

2'. The amplitude - frequency charac teristic of the system   was  based  on the simplified, equation of the ship's rolling.

Thus, having Dg and D Q value s of li ne ar approximation obtained from di a grams , we ca lc ul at e by formulas (2) and (4) the st at is ti ca l lineariz ation coeffi cients (ùç and 2*0 e i n t h e   first appro ximation. In spi te of re lat ive simplicity of expressio ns (2 ) and (4) i t i s more re a  sonable to use a programmed microcomputer, because analogous ca lc ul at io ns are ne ces  sary to be ca rr ied out in the next st ep s.

(üx+/ u44)ë +x44 è^DheV

L

 

(6)

-aeeoc0Dh,sirt,öt, where designations are those as generally accepted in the theory of ship motion'.

29

3. Reduction coef ficient

%g for the Froude-ttrylov force was calculated by Bla- . gowetsjenskij method. The trial operation of the proposed method te st it ie e to a suffi cient pre ci sion« simplicity and oomfort in the e s t i  mation of the parameters of both linear and nonlinear rolling. The total calculation time depends on the number of approximations and averages 10 - 20 min.

B

Da

•• D é

 b

t(ô) M W

A44 M44

s ê 'S 8 e

-

NOMENCLATURE -Coefficient of approximating  polynomial - Breadth of ship - Roll angle variance - Roll velocity variance - Variance of wave ordinates - Acceleration of gravity - Initial transverse metacentric height - Significant wave height - Mass moment of inertia - Lever of statioal stability - Mass of ship

- Spectrul of dencity of waves ordinate - Statistical linearization coefficient - Wave slope - Roll reduction coefficient - Damping coefficient - Added mass coefficient - Frequency of free rolling motion - Wave frequency - Mean quantity of roll frequency - Mean quantity of roll period - Roll angle - Mean quantity of roll angle - Roll amplitude of three persent  probability of excess REFERENCES

1.   Blagowets^enskij, S.N., Golodilin, A.N.

"A book of reference static and dynamic of the shipr. v.II, Dynamio of the ship", Pub lishing office "Shipbuilding \, 1975, Le ningrad, 176 pp'. 2.  Gerasimov, A.V., Book» "Energy-statistioal theory of nonlinear irregular ships motion", Publishing of fice "Shipbuilding", I979,  Leningrad, 228 pp'.  y. Golodilin, A'.N., SchmyTyov, A.I., Book: "Seakiping and stability of ships in - 30 -

rough seas", Publishing office "Shipbuil ding"» 1976, Leningrad, 328 pp. 4.  Trohimenko, I.E.,  Ljjubech, F.D. "The radiotechnlcal calculating on microcompu ters", Publishing office "Radio and nioation", 1983, Moskow, 325 pp'. s Dr.   Shestopal, V  . Dr.  Pashchenko, Y'.  Nlkolayev Shipbuilding Institute Prospekt Geroyev ßtalingrada, 9, 327OOI, Nikolayev, USSR.

Third International Conference on Stability of Ships and Ocean Vehicles,  Gdarisk, Sept   1986

BjJ'CO 0 ) (0)\o

Paper 1.6

INCLINATIONS OF A SHIP DUE TO ARISING SEAS

Yu.   Remez, I. Kogan

them is the determination of  capsizing  probability within some period  of naviga tion time under certain hydrometeorological conditions. Por the first time this  problem was formulated  and   solved  by Pirsov. The second problem includes the de termination of capsizing probability of ship being in the above-mentioned hydro meteorological conditions.

 ABSTRACT For the calculations of ship motions  by the methods of stationary random  prooess theory the parameters of wave spect rum corresponding to some force of sea are chosen. But the wave force scale conforms to the wind force scale only for fully arisen seas. Por other stages of seas de velopment such coordination is absent. At .the same time " for the estimation of the danger of capsizing and    elaboration of criteria of stability it is necessary to consider

2.  PROBABILITY OP CAPSIZING DUE TO ARISING SEAS

 According to Pirsov's general ideas, we consider the problem  of capsizing pro  bability of a ship, due to arising seas.  We introduce the assumption that the storm is developed under action of steady wind, having a constant average speed.and   the seas excites only rolling motion. Capsizing is understood as an excess of some dangerous angle of heel due to rolling. This angle 6„ is considered  to  be known.  Application of the hydrodynamic  and spectral theories of motions allows to determine the statistical characteristics of rolling in oblique seas. With solving the problem  of capsizing both the  vari ance and the average period  of  rolling are considered to be known. The time of  growth of the storm is divided into some intervals. Each of them is specified by its duration and    degree of seas development. The latter numerica lly equals the ratio of average speed of wave running to average speed  of wind.  According to experimental data the  value of this ratio lies within the limits of 0.27 and   0.82. Both the parameters of wave spectrum  X  and H 4 v are connected with the average

the simultaneous action of v/ind

and waves, not neglecting the dependence  between them. In the paper the investiga tion of ship's inclinations and   danger of her capsizing under the action of both the wind with constant average speed  and   waves of different development stages up to fully arisen is made. The probability of capsizing within some period  of navigation time under certain conditions and   summa rized probability of oapsizing in given region are obtained. 1. INTRODUCTION The Register of the USSR  and   other classification societies use chiefly the deterministic procedures for  estimation and elaboration of criteria of stability. Ehe probabilistic approach is used only in some certain parts of the calculations. At the same time it is possible that the pro  babilistic approach to estimation of sta  bility will be found perspective. In any case one may use the'probability of capsi zing as a comparative charaoteristlo of safety of different ships. The determina tion of the probability of capsizing of a ship, being in heavy seas, includes  two different problems. The content of one of - 31

speed  V of  wind  end the degree  ft of seas development by the following Tltov's formulas: f<»0,64Vfi>,

K 5 % » 0,0321 V 2JS V2. Having obtained the statistical cha racteristics of motion Deft and   T/eji for every degree of seas development, it is  possible to calculate both the probability of excess of dangerous angle of heel

OJS

exp LJL) V2D J

VP(A)-P(e>e,.T.
ej

and the quantity fl of excess of  this angle with some interval  of time T

Since we suppose that the excess of dangerous angle causes the capsizing of a ship we are interested  in the first  and unique excess only. Squalling tl to one we  obtain the time T 0 in which with the probability P0a the capsizing of ship will occur. The safety of a ship under fixed deg ree of seas development is characterised  by the value of probability of capsizing within the time interval from nought

toTjj

iy(8.^,T.*T,)-i-«p(-^).-Parameter W* is determined from   the condition that the probability of capsiz ing P & equals P  0 A when T a equals T,o • » . « ml..... Thus

1

The following events favour A - eventt 1) A, - the ship capsized  at the first stage of seas development, 2) A, A2. ~ * be e n i P à**1 no* capsize at the first stage of seas development,  but capsized, at the second stage, 3) Ä , A  2 A 5 - the ship did not cap size at the first and second stages of seas development, but capsized  at the third stage, etc. Thus, it is evident that

P(A)-P(A,)+P(ÂA>P(ÀÀA5)+„ or t-i

 m

PA-Pp.+rPj.inCi-PfO, «3) J 

i-2 J  j-i * where fTl is the quantity of seas develop  ment stages. ' The solution of the first problem is terminated  by formula (3) given the proba  bility of ship capsizing during the whole storm duration: .

•

•

•

'

•

m

.

p

r-Wi-, »)

Further, knowing the repetition of occurence of steady wind having the ave rage speed  V k , it is possible to calcu late the summarized probability of ship capsizing during the whole duration of her navigation in the given region:

and

If P 0 * is sufficiently small, the latter expression will be as follows

p - p M  '.

Further we shall obtain the probabili ty of capsizing of a ship being in  con secutive order under action of all.stages of seas by the wind v/ith constant average speed. Let A'% be a random event consist ing in the capsizing of a ship during i-th stage of seas development while  A is a random event being the capsizing during all the storm duration. The  probability "at  of Ai event is determined  by the formulas (1) or (2). It is necessary to obtain the probability of an event A de noted by P A t

M

Pc=i;^.

(2) -.32

«hers the probabilities P ^  mined by formula (3).

are deter

3. RESULT OP CALCULATIONS  Applied the obtained formulas we have executed the calculations for the ship ha ving the length 45 n», displacement 357 tons and metacentric height 1 m in a three -dimensional beam seas. The results of calculations of average amplitudes and average periods of rolling depending on the degree of seas development and average speed of wind are shown in Fig.1. The pro  bability of excess of different dangerous angles of heel depending on the average speed of wind is represented in Fig.2.

each average wind speed conforms to the  most dangerous particular degree of seas development. It confirms the necessity of rigorous accounting all the stages of seas development when the summarized probabi lity of ship capsizing in storm, duration is calculated.

8°;Te,S 15 -

2ty^~~^

10

.

JL

V» 30 m/s

V

~ ' 

5 •

l\l25_" •

V, m/s

20

Fig.2. Probability of excess of , different hoel angles i

 _ ,

_

0.3

—

i

—

0.4

;

_ _ i — .

0.5

1

0.6

:

-i 

0.7

Fig.1. Average amplitudes and average periods of rolling in arising seas

.

 Ï

1

Having obtained the value of dange rous heel angle and knowing from damage statistics the magnitude of critical cap sizing probability, it is possible, using the graph like Fig.2, to make the conclu sion if the wind and corresponding seas are safe for the ship or they are able to oapaize her. HOMENOLATURE

One should note the following circum  V stance. At the initial stages of seas de 9o velopment greater average amplitudes of rolling are caused by greater speed of wind; in fully arisen seas the dependehoe Vi is of the oontrary character. Besides, - 33 -

H

average speed of wind» dangerous heel angle (angle of capsizing): average period of wave spectrum; significant wave height;

Hs^"" 1,32 H,/3  - ware height of three  percent probability of excess; •U 9 r variance of rolling; If g - average period of rolling; Ta - time of storm duration; - degree of seas development;  K  - time of Û - degree seas existanoe; C - average speed of wave running; DgA - variance of rolling on jb - degree seas; T'eft - average period of rolling on ft degree seas; *Oft - probability of excess of dangerous heel angle on & - degree seas; T - arbitrary time interval; rt - quantity of excess of dangerous heel angle within T - time;  [Q - time in which with the probability Po6 the capsizing of a ship will occur; P A  - probability of capsizing within the time interval from nought

A

l\'%

*° V

-random event consisting in the oâpsizing during t -th stage of  _ seas development; 'Ay - random event contrary to Ai;  A - random event consisting in the capsizing during the whole storm; P. '-• probability of A - event; P : - probability of capsizing during i -th stage of seas development; *'  M - symbol of product; 1Tt - quantity of seas development stages; Q. - repetition of occurence of steady wind, having the average speed V^; P - summarized probability of capsiz ing in the given region. REFERENCES I.Titov L.P. Wind wave, I. Gidrometeoizdat, 1969. Yu. Remèz, Prof., So. Dr. J. Kogan, Dipl* Eng. Hikolayev Shipbuilding Institute, Prospect Qeroev Stalingrada, 9, 327001, Hikolayev, USSR.

- 34 -

Third International Conference on Stability of Ships and Ocean Vehicles,  Gdarisk.Sept   1986

mmw, Paper 1.7

ON THE FLOATING DOCK'S DYNAMICAL BEHAVIOUR UNDER WIND SQUALL IN A   SEAWAY

Yu.   Bilyansky, L. Dykhta, V, Kozlyakov

 ABSTRACT In order to estimate a dock load car rying capacity, to design its  reliable  mooring system, to carry out well-founded strength calculations  etc.  under wellknown assumption of the hydrodynamical ship-motion theory and the theory of fle xible heavy thread, a study is made of the floating dock oscillatory motions excited  by wind squall and incident waves in anohorage. The dook, anchor cables and   sur rounding fluid  are considered  as interac ting members of a single mechanical system with disturbed  and lumped parameters. Computed numerical results for one environment condition combination are pre sented  as plots both of time functions and point speotra for the dock sway-, heave- and   roll-motion. INTRODUCTION The up-to-date state of the hydrodynamioal ship-motion theory and the modern high-speed computers capabilities make it  possible to formulate and to  solve the complicated engineering problems of con siderable importance from viewpoint of ap  plying the results obtained  in design and operation practioe of such expensive floa ting structures as  offshore platforms, floating docks, drilling ships etc. Unlike an ordinary ship the charac teristic feature of the mentioned struc tures is availability of M(M^)  anchor cables influencing essentially on its me chanical properties, therefore for rea sonable information on  structure beha viour under heavy environment conditions to be received it is neoessary to account the interaction between surrounding wa••'

•

.'•'••••-.35

ter, structure body and anchor cables. Taking this interaction into conside ration complicates, to some degree,  the  problem  on structure behaviour by  making it non-linear, the oomputer-generated ana lysis being appropriate only. In this paper there are outlined some  points of the problem  on definition of the floating dock oscillatory motion in ancho rage under wind squall and  incident waves having been solved  to estimate the opera tional possibilities and to carry out some design calculations for the real structure in operation. 1. EQUATIONS OP DOCK MOVEMENT The dynamical behaviour of the consi dered mechanical system being affected  by exciting forces due to wind squall and in cident waves is investigated under  the following assumptions: « the dock is a rigid body with  six degrees of freedom  and its submerged  vo lume forms a parallelepiped; '...-- the surrounding water is incompres sible inviscid fluid under gravity and its disturbed motion is supposed to be descri  bed in the scope of the linear wave the ory; - each of the anchor cables is homo geneous flexible heavy thread which ten sile deformation is sufficiently small to use Hooke's law; - the only category of the hydrodyna  mic disturbed forces the cable being acted upon is considered» namely, those associa' ted with oable's added mass and   applied in the cable normal plane; - the time scales of dynamioal pro cesses caused  by squall and   waves are in commensurable (the wind-squall dock vibra-

tions' period considerably exceeds that of waves); - the inoident waves are two-dimen sional regular waves of small amplitude-* length ratio; - the wave-excited displacements, ve locities and   aooelerations of the system under consideration are first-order small quantities, their squares and    produots  being neglected; - the squall is  simulated  by  time step-function with a known wind-velocity's jump. Owing to the above mentioned incom  mensurability in time scales of the system oscillatory motions due to  squall  and waves it is  permitable to use  separation  principle for the system motions and to subdivide the undertaken study into two some easier problems: 1) the  non-linear  problem    on the  determination of  "slow" dook movements and  quasi-static stresses  by squall; 2) the in anchor cables foroed  linear problem    on the description of the system's "fast" vibrations about its chan ging equilibrium position and   dynamical tensions in the cables' cross-sections.

with the added mass matrix elements  being calculated   by strip-method based  on the solution of boundary-value problem  ön the determination fluid motion disturbed  by a uniformly moving single reotangular contour (whenk^>T) or a grill of such contours (when h "° T ) • There are not any prinoiple difficul  on determination of ties in the  problem  the anohor cable tension Tj   which may be obtained  by  means of the equations well known in the flexible heavy thread mecha nics. In particular, the quantities 6; ,  Vj may be found from   the relationships

»j*kAJ-Aj(iteJGJLJXLÎ-zr)t*.s. " (2)

26j  provided that inequality

For mathematical formulation and so lution of these problems two Ourtesian co ordinate systems are usedi 1)X,Y» 2  in space, X > Y . -plane system  is  fixed  coinciding with basin bottom and 2 -axis vertically upwarf; 2) X, y, £   -system is fixed in the dook, its origin being in the intersection point of the dock central longitudinal plane, midstation plane and waterline plane, the latter is considered as X , y -plane with X - and V -axis direc ted  to the dook bow and to the port res  pectively, while 2  -axis is directed  up ward. When waves and a wind vanishes, the 2 -a nd 2 -axis are coinciding with  the same vertioal line and other ones are mu tually parallel and   unidirectional. Hereafter the  nomenclature tion 4 will be adopted.

is satisfied. This inequality means that angle of tangent to the anchor oable in  is  non-zero; its low end    with the ground  in the opposite case the quantities 6j t  Vj and Lj are the  solutions of equations

à

. O

J

-

'

•

in Sec where

The system of differential equations for determination dock movement due to squall may be  written in matrix form  as follows:

Tj^Gj'stftOêj ;' Tia-BjCQSäj j T i3 -V3 ; "Ijs-T*(Hj -V-£0}-Tj» (Xj - ^o); 

- 36 -

(4)

The wind loads on the doek are to be oalóulated  by using the formulae

of the cable's steady-state oscillations   of the  dynamical stresses in its and  cross-sections  (the prime is used   to de note the differentiation on  dimensionless curvilinear cable's coordinate &  , the  £• *• 1 corresponding respective £ » Q and  ly the lower and  upper cable ends).  In. order to account- dynamical interaction  between an anohor cable and the dock,  the vector U. with components U 1 , U 2 , U 3 is to be satisfied  to boundary conditions.

uCo)-0; 

(5)

Fas ** £ a ^ m - x a F « *» Fa6 *" *a fq.2. ~ ^a 'ai •

 Written formulae (1) - (5) form   the olosed system  of non-linear equations in respect to the unknown dook displacements and  to the anchor cables tensions due to squall, the solution being obtained  by nu  merical method only.  Assuming time dependance of  unknown kinematio and dynamio parameters defined  by factor 8 '®" (the real part is only to  be taken into aooount in expressions con taining this factor),  it is possible to reduce the description of considered  mechanloal system's small vibrations about its equilibrium position to the  calcula tion of their complex amplitudes. Before writing the dook motion equa tions it is necessary to consider question on the anchor cable reaotion applied  to the dock for its complex amplitude to rep resent as a linear combination  of  those of dock motion. The following system  of ordinary dif ferential equations

•^tt1+-Q;,--*Qti;-:u^.-*uI-; oau2 + Qi-:-^i-. :-.;

is found to define the complex

when £ «» 0 and £ •= i. respectively. It should  be stressed here, that  W j is the dock's ^j.-th hawse velocity vector refer red  to the cable moving trihedral. As it  may be shown by methods of the  analytical theory for differential equations the so lution of the eystem  ( G ) ie represented  by power series £l}. On the^ basis of the  computations, which are simple but some long, it may be found  the complex amplitudes' column vec tor ri of the dynamical cable tensions acted upon dock in the j -th hawse point to be represented by

i

amplitudes

Bi R

where elements of matrix D< are dependant on geometrio parameters of j -th  anohor cable, on the vibration frequency  and on thé mutual dock and cable arrangement. The system  of six algebraic equations for determination of the dock motion com  plex amplitudes may be  presented  in form of matrix equality

•£M i*l 

(6)

u(i)-Wj

R

(7)

J

where the dock damping coefficients matrix X and  added mass elements matrix   U are to be calculated  by strip-method based on the solution of boundary-value problem  on determination of the finite depth fluid's  motion disturbed  by a rectangular oontour oscillating on the fluid free surfaoe. To make the paper shorter it  should - 37 -

 be only noted that the latter problem may  be shown by ship-motion theory £3] conven tional methods to reduce to the solution of an infinite algebraic system with  res  pect to the Fourier series expansion coef ficients of the hydrodynamioal singulari ties unknown density to be disturbed on the contour.

taken for all j «1,2,...,12t 3). the anchor cables material Young's modulus is 11 9 adopted equal to 1.8 x 10 N/m . Table 2. Dock data

Length Breadth Draught Mass Mass moment of inertia

The limits of the presented approach applicability is mainly defined by the va lidity of the assumption that the time scales of dynamical processes caused by squall and waves are to be incommensur able, the mentioned property of the system under consideration being dependant upon the anchor system stiffness (defined by lengthes of the anchor cables provided the water depth is  constant)! for the "soft" anchor system characteristic the incommen surability is available, for the "hard" one it is absent. In the latter case the study under taken may be made on the basiB of system (1) which is to be defined more exactly by introducing exciting forces due to waves, damping ooeffioients matrix & and by changing the added mass matrix M 0 on that

275 m 63 m 5.3 m 8.86x10 7 kg

Ix £»«

£>s*

£U4 £i24

X»

X33

A 4« Zc

Coordinate of CB Coordinate of CG Waterplane area Waterplane area moment of inertia Above-water centerplane projected area Coordinate of

E

9

3

3.77x10 1 0 kg 4.43x10 7 kg 2.66x10 8 kg 3.77x10 1 0  kgra2 2.1x10 8  kgm 1.53x10 7  kg/seo 9.18x10 7  kg/seo 9x10 8 kgm 2 /seo -2.65 m 4.7 m 1.84x10 4 m2

ox

7.05x10 é m4

Sa

7.8x1O3 m2 8.5 m

2a

Table 3. Anohor system data

In this Section there are represented some illustrative results on the osoillatory motion of the floating dook having  been built by a Yugoslavian shipyard and established on the water area of Novoros siisk snip-repair plant. One of the most interesting, from praotlcal viewpoint, cases of environment conditions under beam waves and wind is considered, the input waves, dock and anchor system data being  presented in Tables 1, 2 and 3.

%

1 2 3 4 5 6 7 8 9 10 11 12

Table 1. Wave data

In addition to the information con tained in Table 3 it is important to state the following» 1) in the Table the an chors' coordinates in X , Y , B -system are written assuming the wind's absence; 2) for the anchor cable unit length mass the same value ttlj «. 238 kg/m have to be

tn

Damping coefficients

2. RESULTS

* 0 93 m 3.7 m 0.65 sec"' •W 0.0676 m"1 k 11.2m

B T

Added mass coefficients

of £1 .

 Wave length  Wave height Frequenoy  Wave number  Water depth

L

Li m. 88 88 88 88 88 88 88 88 146 145 146 145

XAJ \

m m

87 82 25 27 -25 r27 -87 -82 30 30 -30 -30

X

i m

54 '•' 87 82 -54 18 54 -5 4 27 -18 54 -54 -27 54 -87 -82 -54 64 133 64 133 64 -133 -64 -131

«i m -32 32 -32 32 -32 32 -32 32 -3 7 -37 -37 37

Goj KN 220 220 220 220 220 220 220 220 350 350 350 350

On the basis of computer-generated numerical results for dook motion in Fi gure 1 there are represented the dook  dis  placements under swaying, heaving and rol ling motion as à funotion of time and in Figure 2 there are given the mentioned  processes' point spectra normalized by the  maximal speotra ordinates. The availabi lity of anohor system, as seen from Fi gures 1, 2, affects weakly dock heaving 38 -

"Ij. STj»*} *., - oolumn vector cable loads acting on dook under squall;  A, x, Z - fixed  in space coordinate system; U « {U-^ly.., - thread vibration amplitudes vector referred to moving tri hedral; - cable vertical tension compo  Vi nent in hawse point; - wind velocity;  V  - hawse velocity referred  to  W; cable moving trihedral; - hawse point coordinates in Y« V. 7> X, x,r -system; XA I , YAJ  - anchor coordinates X, v», e fixed in dock coordinate system; Xj, y4- Zj-«-T - hawse point coordinates in x, y, a -system; ~ coordinates of centre wind Xa,y>,3» 1 - dook damping coeffici ents matrix; ^,0 - wave length; J_. ^ f - auxiliary variables in (2) and

notion and influences greatly its swaying and rolling motions; the latter makes vi   brations more complicated and their spec tra includes some additional harmoniesbe sides main ones. 3. NOMENCLATURE

B B

- dook breadth; - matrix of coefficients for rep resentation  of the cable dyna mioal reaotion; - restoring foroe ooeffioient ma C trix; - airforce coefficient; Ca - anchor cable elongation; G; j^alCj,^ 6 . - column vector of airforoes due to wind squall; - cable horizontal tension compo Gj nent; - initial cable tension; Goj - gravitational acceleration; 0 - wave height ; H - water depth; fl ~ dook mass moment of inertia; ly - waterplane area moment of  iher3n - subscript for referring to a qua ntity associated with j' -th an chor; 1c » - wave number; - dook length; L - anchor cable length; Lj - saging part of anchor cable; 4 - projection of Lj on the bottom; •^n vector of dock motion displacements; 1

 At « ^ U p s \ P 8m1 - dock added mass matrix under motion;  M o m  {^ors J*-,s-i ~ dock added mass mat rix under squall motion; v"{"*\»}r-i - unit vector of the wind force principal veotor; - air density; OQ, 6*  '..'•- oircular frequency;


 motion displacements; £ 0 « 0 £ 0 - dock displacements along X- , y -axis, respectively, due to squall; - dock roll-, pitch- and yaw-mortion displacements; do "QiT0)rO~ °fc angular displacements due to squall.  dot denotes differentiation with  A  respect to time.

V  téo.ioA, e0,^0> fa}-column v6otor s

Sa T

veotor of dock displacements due» to squall; - dummy indices; - waterplane area; - dook above-water centerplane  projected area; -• dock draught;

K ~ dock surge-, sway- and heave-

39

SWAY t

SWAY

 z——~

0.5

-2 —'—  — O

20

40

feO

SO

IOO  sec

HEAVE O

II..

0.23

Il .1 O.SO

hol

0.75

4.

11

aa£'

HEAVE

05

ROLL O

O.S 5

0. 50

il llll. 0.75

1 II 1. sec 1

0.75

1.  sotf'

K01L 1.

0.5

O

20

40

60

ÔO

100

O

sec

Figure 1. The dook oscillations under motion

0. 25

O.9O

,l

Figure 2. The dook oscillations point speotra

- 40 -

I,

 A prime denotes differentiation with respect to dimentionless variable £ .

4. REFERENCES 1. Bilyansky ïu., Dykhta L. Determination of the small vibration forma for a heavy thread in ideal fluid. Hydrodynamics of ship. Sbornik nauchnykh trudov, Nikolayèv Shipbuilding Institute, 1984, pp.84-91. 2. Qurevicb M. Added masses for a grill of rectangulars, Frlkladnaya matematika i mekhanika, vol.4, No.2, 1940, pp.93-100. 3. Haskind M. The hydrodynamical ship-mo tion theory. »Sauka», 1973, pp.1-327.

ïu . Bilyansky, Dipl. Bng. L. Dykhta, Dr.  V. Kozlyakov, Prof., Sc. Dr.  Nikolayèv Shipbuilding Institute , Prospect Geroev Stalingrada, 9« 327001, Nikolayèv, USSH.

/

Third International Conference on Stability of Ships and Ocean Vehicles, Gdarisk, Sept. 1986

mmm,

Paper 1.8

THE PREDICTION OF LONG-TERM SHIP ROLLING FOR INTACT STABILITY AND ANTI-ROLLING SYSTEM ASSESSMENT  by

Y.S.  Tao"

 Abstract

In this paper it is suggested that ship roll performance in irregular waves, including vess els with anti-rolling systems, ought to use long-term predictions of roll motion using an accumulation of the shortterm ro ll response with a cond itio nal pr oba bil it y. In th is approach the ship ro ll response for short-term pr ed ic ti on i s cal cu la te d by a non -li nea r method. The author s have analys ed a number of l ong-ter m pr edi ct io n methods of roll and suggest that the roll damping can be given by means of progressively approximate methods corresponding to si gn if ic an t ro ll angle which i s cal cu lat ed from the ro ll response amplitude oper ator. It is suggested that the rules for stability of sea-going ships should adopt long-term predictions o f extreme value in roll angle as a basis for ship st ab il it y cr it er ia . 1. 

INTRODUCTION

The rolling motion of a ship In irregular wave is a Important Index for dynamic stability and seakeeplng. Although a comparison of rolling performance between ships of different designs can be made from ship notion prediction programs or from modal tests these are normally based on short' term predictions in irregular waves. If short term prediction techniques are used for roll prediction the comparison between different ship designs can only be made for. sp e c if ic parameters. However, a more meaningful comparison can be made if the variables of loading condition, speed, heading, sea state etc are combined with the encounter probabilities of the factor s involved over , the ship 's ent ire li fe ti me .. . ' - . ' " ' •

Ship Hydrodynamics Laboratory, Shanghai Jiao Tong University, China •

- 43 -

So as would be expected in a number of Intact stability rules for some existing sea-going ships stability criteria for the roll angle is based on short-term rather than long prediction. The calculation of roll response amplitude opera tor i s usu al ly based on lin ea r .theo ry although rolling is finely tuned phenomena and is highly non-linear as a result non-linear damping and non-linear restoring moment. However, i t i s now pr ac t ic a l to make use of non-linear prediction techniques using the equi vale nt li ne ar is at io n t e c h ni q u e. , the perturbation method, the FokXar-PlancK equation method and the functional representation method for the prediction of non-linear ship roll motion. It is therefore possible to calculate a vessel's rolling performance in irregular waves, and assess the effectiveness of a vessel with an anti-rolling system, using non-linear methods and long-t erm pr ed ic ti on tec hniq ues. The rolling motion obtained in this way is an accumulation of the short-term roll motion obtained, by the con dit ion al pr oba bi li ty . The roll damping can be estimated by means of an approximate method baaed on the si gn if ic an t ro ll angl e ca lc ul at ed from the ro ll response amplitude op er at or / The . cr it er ia for the

stability of sea-going ship» can then formulated using long-term predictions of extreme value of roll amplitude.

2.

bo the

T/3NO-TKRH ROLL ANGLE .AMPLITUDE PREDICTION METHODS

It Is of considerable interest to assess the magnitude or roll angle amplitude experienced by a ship in her lif eti me while the ship is s t i l l in the desi gn st ag e. This i s achie ved by evaluating the long-term rolling response of a vessel which is i.e. essentially the application

EEEE

P0t,(«>*l) •

 ££Z

E " e 

of the conditional probability to the short-term response. In order to evaluate roll amplitude the lifetime of a ship in a seaway, the amplitude of roll for various short-terms are computed and are accumulated taking into account the frequency of occurrence of each short-term. Because the amplitude of a roll In the short-term follows the Rayleigh probability law, and the sea condition can bo represented by the significant wave height and wave period modal, then the probability that the amplitude of a roll, «, will exceed a specified tfi in the lifetime of the ship can be given [1] as

•»
E E " • p
W H T0 V X

a is  standard deviation of   roll amplitude  (short-term), X  Is heading to waves, V is ship speed, H is  significant wave height, T 0 is modal period, w is  loading,  and n* is the average number  of  zero-crossings  per unit time for each steady-state short-term response.   it. is obtained  as  follows  Where,

° 2TC

•y m z/  m o

< 2 >.

where, m0  is area under the short-term re spo nse spectrum, mj i s second moment of the short-term response spectrum. 2. 1

term tests, and show very large data dispersion, and Insufficient to be dealt st at is ti ca ll y. It is gratifying that Takahashl 1*1 carried out full scale measurements on a container carrier, fo r abo ut 52 months from Oct obe r 1976 to January 1981. The cumulative probability of maximum, minimum and RMS va lu es of r o i l amp lit ude are shown in Fi g. 1. The ra ti o of the frequency of wave encounter angle for this vessel was approximately as follows Head i Bow i Beam i Quar ter i fo ll ow = 2 i 3 I 2 i 2 i 1. ,

Fu ll sc al e Ship Measurement Method

The shi p's r oi l motion in seaway i s ef fe ct ed by very many environmental and operational factors, and it is not possible to Include all; of these in Borne of which have not or cannot been considered practical studies of ship motion. Furthermore It is not practical to collect long term data from ship trials talcing into account all environmental and operational conditions a vessel may encounter during her lif eti me . In the in it ia l ship design stage i t is very useful to assess the motion and stability of a ship to Use an analytical expression from the probability distribution of ro ll motion based on fu ll -s ca le dat a. Although there have been very many roil full scale tests for many kinds of ships, most of them were short

X

Fi g. l

i f

r  i

'T.

1

• J.

I

\ 

i

i-JL. i i i .i

4 ^5 « 7 • »10

r

U 

•

.

i. •

20 »

Rolling Angle (dig) cumulative pro babi lity of rol lin g angle

2. 2 . Rob ert s e t a l . Method •/'•.. Roberts e t a l . hold l' l . that In the most general case for specific ship, long-term the standard deviation of roll, oi. Is a function of four vari able s - i . e . H, T 0 , x and v, as

Oj,'* UB, T„, X. V)

O) •

to significant wave height is constant, d. in a specific apparent wave period zone, l.e

and these four variables will De governed by à. four-dimensional joint density function P(H, T0 , ,

 X,  V). if x and v are assumed to be mutually independent, and also independent of h and T0 thus

d » -Ja R4/H (7)  Where, d is function for mean apparent wave  period, 10 .

Pt(H, To. X, V) •» PtfH. T 0 )P!XX)PIXV) 

(4)

test.

which can be

ship

full

scale

found by test

ship mode

or

analytical

calculation methods. Rg is value mean square of

where, Pj,(H, T 0 ) is Joint density function fo r H and T0 ,  VxAx)  is density function for x, *i,( w ) *• *ten»ity function for V.

ship

roll

height.

amplitude, And

H

is

assumption

a

significant

wave

specific shipping

route expected maximum value of f0, as shown in Pig. 2, have also been known.

To evalu ate the dis tr ibu ti on of o, a particular case, ship speed V, which is not a random va ri ab le . The heading to waves, x.  has a known probability distribution, one can then approximate by considering a number, n, of descrete headings, such as X\, Xz. . . . . . Xn-

w.«« r*r*ed oifttrttetiaa NT.)

The final density function for a  is then obt aine d by summing ove r a l l the head ing s.  Ptfo)

- $

Thus

£   P(o|jfi) i«l

<5 > It Js shown, the cumulative probability ' distribution of  a. we can regard o as a j oi nt : de ns it y func tion of H and T 0  and a simple numerical Integration technique has been proposed from Roberts etal which can be used to c o m pu t e the cor resp ondin g pr ob ab il it y distribution óf p.

Fig .2

Dis tri but ion of di Function

Based on the assumption wave amplitude long-term probability which is welbull dis tr ibu tio n for al l ÎQ value, the ship ro ll amplitude long-term probability distribution can be gi ve n by .

Finally, a long-term probability that the standard deviation; o, will exceed a specified value, o«, we can be found by

Pt,(*a) - > - exp( -   P ^ o o , ) =» £

Te xp {- <£j )

i-^-y]

| . Pi(H, T0 ) • . ' " '

(8) <

6

>

.

where; o ,is standard deviation for a : specific loading; a specific ship speed, a specific heading to wave and wave spectrum value , of a specific H arid T0. Ci and  ax are a specific welbull parameter that the simplest case could be constant values of equation (6).

where, a, b and c are weibull parameter, d is ratio value which can be given by equation (7). Finally, roll

long-term

probability

amplitude, : tfa- will

exceed

that a

ship

specific

value, $ai> one can be found by Pl.(*a>*a,) "». fPi(T 0 ) . exp / - ( *° l 4=1 Bi ll ig

)C \ /

2.3- Hordenströn Method - •{»)  A

simple

and

convenient

approximately

calculation have been,given by Nordènstron C*l.  When

long-term

probability

distribution

of

wave height and wave period are not too exactly known, we can be according to- Nordenströn idea for ship wave induced bending moment, assumption the 'ratio of significant roil amplitude (double)

where, Pj,(f0) is ship encounter wave period  probability for i set, cj., bj, and ci are ship roll amplitude Welbull distribution parameter for 1 set of wave, period, d   A  is ratio of ship significant roll amplitude to significant for i set of wave period.

 We can be rapidly and convenient estimated i ship toil amplitude long-term excess probability

probability. (3 ) Al l above methods for esti mate d ro ll amplitude long-term cumulative probability is not too larg ely diff er ent , except Roberta et al method, which mainly has lar ge dif fe ren ce s in wave parameters between Roberts* calculation and Takahashl full-scale measurements on ship shipping route.

Crom equation  (9),  which is only wanted wave  period long-term probability ship shipping route. 2.4

distribution

on '.

Wave Height Replacement Method In

Pig. 3,

observed

wave

cumulative

height

have

probability been

of

illustrated

during Takahashl full-scale measurements.

The

results shows that RMS of roll amplitude is very nearly with' observed wave height but have need to exchange metre of wave height unit for degree of roll amplitude unit, or opposed exchange them.

0.11

o.t . o.i 6.1 o.« 0.1 o.t )

»

totting

Pig. 3

s

»

i • »to•

IU19U t««9 )

•

i»

ii

ii

, v t n Mig ht *••

cumulative proba bili ty of ro ll in g angle and observed wave height

Fi g. 3 shows simple and conve nien t es ti ma tes of RMS ro ll amplitude f or long-te rm cumula tive probability from a ship on a shipping route with observed wave height (or significant wave hei gh t). Then we can obtain si gni fic ant ro ll amplitude of. long-term probability and the expected extreme of long-term roll amplitude. 2. 3

va ri ou s Method Comparisons

In order to compare the above various methods for estimation ship roll amplitude long-term pro bab il ity dis tr ib ut io n characteristic, thé results of long-term probability of four methods have been shown In F lg .-

4.

• ' " _ • . . •

The Pig . 4 shows, ( 1) r o l l amplitude long-term probabil ity d is tri but io n i s cl os e , to that weitoull distribution. (2 ) Roll amplitude long-term pro ba bil ity Is as good as 2-paramntor weibUU distribution, tha t Is , the Ray le igh type. Theref ore, so long as parameter la selected good for various co ndi ti ons that Ray le ig h dist ribut ion' .." is suitable to predict roll amplitude long-rterm

0 ftf'9 I

Pig .4

2.6

tong-term proba bili ty di st ri but io n for o

Expected Maximum Peak Val ue

Por eva lua ti ng the long-t erm expected maximum peak value probability distribution, one can ev a l ua t e ex pe ct ed maximum peak valu e frequency of occurrence of each , short-term for various spa state, ship speed, loading and heading to wave e t c . ; By means of exa ctl y the same way as in the treatment of  : a  or RMS, one could determine the di st ri bu ti on of - expected maximum peak v alue, E(Aniu) • Based on Takahashl full-scale measurements data, one can derive the relationship.between ' the . long-term di st ri bu ti on of long-term distribution of o or RMS.

E(A m ( l )

and

Prom as shown in Pi g. 1 cu rv es , th e prob abi lit y densi ty function of E (A mM ) can be gi ven by ' •*W*nin))

- K^ P< R *> ;- KS  

P( 0)

(10)

46 -

where, P(R$ )- and . P( o) represent pro bab ili ty, di st ri bu ti on of RMS of . ro l l amplitude and a of roll respectively. On the ass umptio n, tha t r o l l amplitude probability distribution accord with Rayleiqh type, thus coefficient and can be respectively by

2. 7

The pro ba bi li sti c extreme value of ro ll is def ine d as •• the la rg es t val ue of the maxima (amplitude or peak-to-trough excursions) which wi l l occur in a sp eci fi ed number of obse rvat ions or in a specified period of time. Pot th is rea son, the to ta l number of ro l l, N,  experienced by a ship in her lifetime has been proposed by Ochi C11(S) which are arbitrarily in the long-term prediction methods developed to da te . Ochi hol ds that they can be eval uate d from the numerator of equati on ( l ) as follows:

(ID (12)

 K„ *> l . s e - 0.07V o

Where R^ i s th e RMS of r o l l am plit ude, o the standard deviation of roll.

is

The co ef fi ci en t K usu all y i s a functio n of o r o l l or RMS of ro l l amplit ude.

""" E E L E E " ' P(X ' 

V H

W H T0 V X

| '

T

°'

Est im ati on Of The Extreme Value of Ro ll

M

>-»,(To|H)-P(H)-P(W)T.(60) Z

 Where, T is total exposure time of a ship in The  other seas in her   lifetime (in  hours). symbols are meaning same as  equation (1) and "..':''. (2). •;  Roberts etàl has also been given (3] as Return period  N « p(o>oj) (14)

(13)

Moreover, the . hydrodynamic damping as so ci at ed with rolling moiton is usually relatively small. It Is possible for a ship, when operating in a random sea st a te , to ex hi bi t sever e r ol l motion which nature to behave no n- li ne ar . Seve ral analytical methods (6,7,8,9,10,11,12,13,14,15,16 17], have receiv ed some at te nt io n and Is found corresponding re su lt s In the fi el d of ship rolling motion.

where N i s measured In number of sea s t a t e s , P(o>oj) is probability that the a  will exceed a specified value, Oj. 3.1 3.

a n a ly ti ca l

Approach

TP

HQn-UPga,r

sn ip The use of this method reduces the problem to a linear one by replacing the non-linearity by a su it ab le lin ea r term. For example, Vassilapoulos method (6] is given as followsi

Bellica . I n o r de r t o e v a l u a t e l o n g- t er m roll amplitude according to previ ous s ec tio n have, to know sp ec if ie d cond iti on shor t-term standard deviation of roll. In random sëas many ships have a natural ro ll ' frequency of' si mi la r magnitude t o the 'frequencies at which wave energy is dominant.

'*N

-—

H

Equivalent Lin ear isa tio n Techniques

For the case in which both non-linear damping and restoration are present, one solve the resulting second-degree algebraic equation in O0. The re su lt for the po si ti ve square root of that equation can be shown to bet

*

'•

Zil

(15)

Where, o* is the standard dev iat ion s for the N : . non-linear ro ll   '::,/• \ .~' 0

.y

i s the stan dard- devi atio ns for the linear r o l l ••.':•;;.,;''•

--47 -

0

is the ratio of non-linear and   linear damping coefficient

 y

is the ratio of non-linear and   linear . . restoring moment

Ug

is the roll undamping natural frequency

no n- li ne ar it y. This reduces the problem to the so lu ti on of sev er al line ar problems. The method is expected to yield a good approximation when the non-linearity is small.

Equation (1 5) permit the est im at io n of | non-linear coll statistics given the values of linear roll variance, aih 2 , and th ose of the mixture ratios 0 and y,  together with the undamped natural roll frequency Ug. 3. 2

Yamanouchl Investigated the effect of quadratic damping oh the roll spectrum on the ba si s of a per tur bat ion method (9 ). one can be given as followsi Accordingly the roll spectrum s 4 ,0,(w) of the 1st approximation, 4>i(t), it were calculated by this author, is taken as

Pertu rbat ion Methods

Here the solution Is expressed In a series exp an sio n In powerB of a ïm a ll parameter, usually related to the magnitude of the

s

0 l « l < w > - s * o » o < w > - 2  ^n °«o °

8

u

*o*o(u>

Vu|

• **lY«>|"(if °i 0 * »  ••„•„(«•o •  J^IL  »<«>l (16) Where, s

s^ .* (CJ) i s the li ne ar ro ll in g spectrum,

qg(ü.) - *m
+ 

^tj,

* o * o < u ) " | H < U > | 2 SCC

 / Ho(0)) t^p

U  t Z 

(18)

7(
( Hq(W) q1 W«2 y(U)\

Ha((J) =

• 3 {{
5

—5

1

" t . Ho(U)

(19)

<17)

S(u)

"  f

f

Sio^^i)

• s «o*o < < *' 2 ) •

The random respon se of r ol l with no n- lin ear restoring function or non-linear damping have also been obtained by Crandall applying the classical perturbation method (11]. If the wave is a stationary random process w i t h known s t a t i s t i c a l pr ope rt ies , the non- line ar restoring function g( 0) - * . we consider only the first-order perturbation, and take the case of Gaussian white ex ci ta ti on , the simple form for the approximate mean' square value, o$ N , of the non-line ar, ro ll can be given by 2

2

«

°«l. - 3 e o * L

(21)

where, 0

*t,

is mean square value of the linear roll,

- 48

io*o(' J -w,-
e

and S t t ( u ) and  Sfofoiu) are wave and li ne ar rolling velocity spectral density respectively. non- line ar Thus,  one can be ca lc ul at ed rolling variance from Integrating equation while ship linear roll characteristics and non-linear rolling damping have been given.

s

(20)

is the parameter for the non-linear restoring function.

For the ship with small, but non-linear, roll damping. Is given '*N

'•L

2 4 3OKJ0 O 0(j (22)

for the first order perturbation approximation 'to the mean square of roll with non-linear roll damping. Where, a  is damping coefficient of non-linear roll. 3. 3

Fokker -plane k Equatio n Methods

Robest [12] demonstrated that a suitable theory could be constructed provided that the roll motion can be modelled as a single degree of freedom eq ua tio n. It was shown, by combining Markov process theory with an averaging technique, that the energy envelope of the roll mot ion cou ld be ap pr ox ima te d as a one-dimensional markov process, governed by an ap pr op ri at e Fokker-Plarick eq uat ion . The

stationary solution of   this equation yields the,

the   roll amplitude. An  'advantage of   this approach Is -that the  non-linear components . of the  damping and restoring moment terms.  In the   equation of '

 pr ob ab il it y

 motio n,

funct ion  of

de ns it y

need  not be   small

and

can be of

arbitrary form.

«•

of   roll response,

Waye

Long-term

Pr obabil ity

DAstrlbMtAQn  As

previo us

mentioned  on  long-term

roll

one of it  should  be  known the  ship

 prediction, sea way

The standard deviation, a,

shipping'

route

lon g-t erm

wave

characteristics.

*(t), is given by

'

Bist; Imqt, ion—Of

The various sea zones or  shipping route wave

. «» O* « [ p(V ) D?(V)dV    ' o

data  can (23)

.Where

bo  estimated  by  long-term

statistic

results which are given as  follows: 4.1

Hogben

-  Lumb

  Walden And 

wave

Observed

 Values

 V - f + U(*>

 Where

(25)

which tables are   presented have been arranged Into SO  groups,  but the North Pacific are not covered.

2 & /2 is the kinetic energy and

C( f )df 

U( « ) "»

(26)

'o Is the potential energy. The stationary density function can' be  given by v

P(v) - ($71 «KP ( a -ƒ ' #f t <«} 

(a')

(28)

°

4.2

North Sea  Around  And off Northern wave -Probability

Norway

  Hogben . t*°) b ase< j 0 n  North  Sea and  around observed data have been developed  a  joint  probability of  wave height and    period  and the corresponding

It is evident that, one can calculate the variance of non-linear roll based on equation (23). But a modified theory has also been applied by Roberts. The idea is to replace S x(u) in drift and diffusion coefficients by a  modified spectrumi

s' x(  u) .=» TtQ(V)] S x (w)

walden t,9)  based  on  nine weather ships in the North Atlantic Sea  zones have also been given the   wave height and  period observed values. The whole year wave frequency have been tabulated  in Table 1.

Pang

 Where C Is a normalisation constant such thati f >'(V) W » l

Hogben -  Lumb ('81  according to  altogether about, 1 million sets of  observations made in years 1953-1961 have been used  to  produce  the wave data tables. Sea  areas in great detail  most  o£ the shipping routes  of the  world    for

(29)

Where •yfQ(V)] will depend on the damping function Q(V) and the shape of the input spectr um. Q(V) can be regarded as an ampli tude dependent damping factor. If restoring moment is linearity and Is considered only non-linear damping of roll, one can be used to correc t the sp ect ral le ve l, at each value of v by equation (19).

regression.

In  fact, for  another shipping route can   also been calculated  by  rang -  Hogben equation when wave height data is known. The wave climate off   Northern Norway have  been measured  and    investigated  by  Haver (21' at the Tromsoflaket area  in the years 1977-1991, while the hindcast values cover the  years 19S5-19B1. The long-term variation in the  wave climate is  given  by the  joint probability distribution  of the significant wave height and the spectral peak period. Haver holds that the reason for  choosing the  spectral peak period instead  of   other characteristic periods is  that this period  is   less correlated  to the significant wave height than the other periods. 4.3

North Pacific Ocean wave Probability

Yamanouchi C22 ) based  on  about four hundred thousand sets of  observed wave values for the

Table

1  Wave

Frequency

in the  North

Atlantic (According

to  Waldch's-Dnta)

Wkvo Period  (l«c) 5

 Ï

I

0.75 1.75 1.75 3.75 4.75 5.75 6.75 7.75 6.75 «.75 10.75 11.75 1J.75 13.75 H.75 15.7S

9

7

11

13

20.51 ! 11.75 ! . 77.78 ' 111.08  ' • 31.2« ! 126.41 ! 49.60 > 16.19 ! •"ö.ivi • • • • • . . • . .

Sua v»r  All M»içhII

13

4.57 ! 1.24 ; 0.47 ! 63.08   ^ 17.26 ' 3.39 i 118.31 !  30.34 ; 3.68 ; 93.69 •  33.99 • 3.46 • 44.36! 33.2«! 4.79 ! .o.u • ..iiiiX 17.30 J_   12.89 • 3.1) >_ 0.07 S 1.90 ! 9.90 ! 8.86 ! 3.03  ! 0.0J ' I.J9 •_ ..i:i 7 ..l..hUX. 1.93 •_ ô.ôô ! "l.OT ! 3.35 ! 3.93 ! 1.9« ! 1.34 •_ 0.00 J_ 0.54 }_ 1.36 «  3.36' o.oi S o.ôi ! Ö.1Ö f   O . l l l o.io ! 0.00^ 0.00 • 0.03^  0.08; .?:'!.!.. 0.05 ! Ó.ÓÖ !  'o.u', 0.23 : •   0.07 > 0 , 0 * J L o.oa [_ ! 0.07 ! 0.06 ! o.oo ; 0.00 ' O.OI • o.oi • ! o.oi ;I ! ! ;

in.•» j 345.43 ; 356.73 ; 1)8.59 ;' 39.05 j

17

;

:

5.6) ;

0.92 ;

075 1-75 2.75 J. 75 1.75 5.75 7.75 8.75 9-75 10.75 11.75 12.75 13.75 14.75 75 <

Sum over All Hfl ght

1000.00

values  in ten  years by Japanese shi ps, have al so been given th e  wave frequency  in the  North Pacific. The annual re su lt s  Is as  shown In Table 2 . t h e  F i g . S shows th at cumulative distribution f u n c t i o n o f   wave height from Yamanouchl and  SR163, a l l  period.

• '» the   North Paciftc (:i0" N- 55 ' N. IIO'E-IICW) (Data from  5R 1631  Annua.

6.75

•i »

2.6V

W j « !   frequency

wave He qM(rn) — __

Ml.CO 185.05 69.66 )8.51 33.46 13.50 10.09 6.63 0.40 0.)4 0.4B 0.33 0.11 0.00

I737.49T  061.I

North Pacific ocean by  Japanese ships in ten years were compiled marine meteorological tables. The 163th Research Committee  of the  Japan Ship Research Association t 23 "  observation North Pacific ocean which i s  same observed se a  zone as Yamariouchi, about two m i l l i o n s e t s o f   wave

2

40.64

0.06 : o.oo : 0.60 Ó.77 0.33 ' 0.47 : 0.09 ! 0.56 0.68 • 0.12 • 0.27 1.14 ; 0.0« : 0.39 0.56 • 0.13 • 0.04J 0.59 ! o.o« : 0.03 0.3« •_ 0.04 I 0.04 0.50 : o.o) ! 0.03 0.6« • 0.30 ^ 0.04 0.05 ! 0.02 ! 0.00 0.06 • 0.00 0.06 : u.oi ! 0.03 < 0.01 0.03 ; o.oo : 0.01 0.02 ' 0.01 ' O.OI

whole Y»*r {Cor Al l  Nine Weither Shipt J

Tabla

Sua over  All Periods

Wave 0- S 7  367 22.04) 8.347

2.5)1 0.62«

0.091 0.0)1 0.021 0.012 0.007 0.002

4-1.073

5-7

Period

0.6)9 l.W

7-9 0)15 4.1)5 7.108 1. •» 7 5 2.324

0.365

0.760

0.73" 9.866 9.507

13 0.013

2.190 1.714

i l - i) 0.015 0.6)9 0.8)2 0.941

0.874 O.J32

0.579 0.252

0.281

9 - 11 0.210. 0.9ß5

0.142 0.213 0.0)5 0.021 0.011 0.001

0.J17 0.183

0.199 0.116

0.187 0.113

0.065

0.048 0.030

0.002

o.oo«

0.064 0.039 0.027 0.003 0.004

0.0)8 0.0)5 0.00)

Sum

{ sec)

0.232 O.432 0.359 0.073 O.096 0.053 O.O36 O.034

0.001 0'. 001

0.017 O.OO3 O.OO; 0.001 0.001

Period 8.654

37.920 23.416 14.659 6.178 1.878 0.972 0.724 0.260

0.159 0.103 0.013

0.001 0.001

0.001 0.001

0.016 0.00} 0.012 0.001 0.001

:  0.001

0.002

0.002

0.004

0.003

0.012

27.053

19.771

6.733

3.7PI

1.659

100.000

- 50 -

0;02T 0.005 0.005

perio d can be des cri bed by the log-normal di str ibu tio n. The long-term joi nt pro bab ilit y of wave he ig ht and pe ri od form ula, the probability density of wave period equation and the pro ba bi li ty den si ty of. wave hei ght for co nd iti on perio d equa tion have been . given respectively in those papers. 4.4

The Family Of Wave Spectra

ochl C26 ' is proposed the design extreme value based on the long-term prediction approach usin g a . family of wave sp ect ra in each sea severity. Therefore, the six-parameter spectra havo been developed by Ochi t Z 7 l. The wave sp ec tr a are decomposed, into two parts - one which Includes primarily the lower frequency components of the wave energy and the second which covers primarily the higher frequency components of the ene rgy . . The en ti re spectrum is expressed by a combination of two sets of three- param eter spectrum. These parameters are

10 15 20 H(m)

Flg. S

Cumulative di st ri bu ti on function of wave height In the North pacific plotted on long-normal probability paper.

significant wave height, modal period and shape parameter. By combining two s e t s of thre e-pa rame ter spectra, one representing the low frequency components and the other the high frequency components of th e wave energy, the foll owi ng six-parameter spectral representation can be derivedi

Mano and kawabe C2«]l*S] studied on the st at is ti ca l characters of wave st at is ti cs in the North Pacific Ocean which la showed that the frequency distribution of both wave height and'

4X, + 1

<-

s w

<>- i

£ —rr*jr


S*i  • '

exp - <

4X

I+

1

X^ff)

(30)

be

minimum capsizing lever (or moment). The problem is in calculation present roll angle, though some Irregular wave characteristics have been concerned, but it is based on the short-term predic ted r ol l angle which without respec t t o the possible maximum roll angle in ship's lifetime.

So far, . the rules for st ab il it y of sea-goin g sh ip s in Japan, USSR, P.R. China and othe r a number of shipping register, while with respect to the stability criterion, it is necessary to prolong symmetrically the curve of dynamical (or st at ic al ) sta bi li ty to wa rd s t h e n eg a t i v e di re cti on of the ro ll angl e, *. Then can be concerned through. th is ro ll angle that Is given

we hold that the reasonable way ought to consider the possible ship loading, ship speed, sea st a te of. encounte r and. heading t o waves In the 'lifetime of the ship, 'I .e . for the st ab il it y cr it er io n in the ru le s; for st a bi li ty of sea-going ships is to be given long-term roll angle. Assessment of shi p gener al r o l l , performance and a n t i - r o l l i n g e f f e c t i v e n e s s • f o r v a r io u s an ti -r ol li ng dev ices , i t als o Is given based on short-term predicted roll angle for special ship conditi ons and sp eci fi c sea sta te so far. It Is

 Where,   j

higher

=

1,2

stands

frequency components,

significant 'wave height,

for

the

lower

respectively.


and H is

frequency

and X is spectrum shape parameter. The families family

parameters are of

for

tabulated

spectra

for

six-parameter in a

Table 3 desired

so sea

spectral that can

a

generated from equation (3 0) . 5.

ASSESSMENT OF SHIP HOU . PERFORMANCE

- 51 -

Table 3

", Most Proti-tblv Spvclnra

•

Va I ui»!, àl   SX-pararnrter«, (m-unils)

,

 \

0 V. H 0 J4„ o ;o .'••••' •* „ l . l ï . - ° - ° » H

3.00

I » , - ' * '

H

0.« H 0. 31 H o.jo . • o o t * H 1 50 ,-°- y *  H

1.3J

2 48 . -°"»

„

0 65 " 0.76 « 0.61 . - 0 0 3 ' H 0.94  .-O- 0 5 * H

4.95

2.48 «

0 l 0 2

II

, .i o. -O ^H

3 00

2 . 7 7 , - ° l l î 

II

II 0.B8 ,-° 0 " M

2 55

1.82 .0.089

H

O.TC « 0.44 II 0.81 . -° °» Il I . 6 0 C - 0 0 " «

ISO

2.95«- 0105 H

H

4.50

1.95 .

O.M H 0.68 H 0.70 .- 0 0 '-* H 0.99 c'° °" «

6.40

1.7« , -

0.92 H 0.39 H 0 . 7 0 , - 0 O 4 4 H

1.37  . - » • « ' H

O.M

H

0i6 0,54 H 0 . 9 ) . - ° H

O.M M 0.54 H 0.41 . - ° m  957.  Confid'jncc Spectra

X

".»

"l

0.77 H O.M H 0.54 e'°

0 M

H

0 0 8 2

I.

0 0 M

II

0.70

l.».-»-0"

||

» H 1,30 e" 0 " 0 » H

2.65

3. 90 « • ' « '

,|

<>•»« H 0.54 H 0.62 c - 0 0 » H 1.03 e " 0 0 5 0 H

2.60

O.SΠ e - °

Il

O.M H 0.54 M 0.74 , '

0 0

0.61

«•'

H - l lg nl f' un c wove lulghc lu crcccr«

evident that should be lead to not correct concJusion; For example, Morenahlldt l 2 8 l  and Williams t 29 ' full-scale snips tests for with various passive anti-rolling tank shows that the ant i-r oll ing effe ctive ness i s very differ ent for the same identical anti-rolling tank in differential ship speed, sea state of encounter and heading to wave. Th ere for e, 14th ITTC Seak eep lng •  Committee recommends a standard measure for the effectiveness of passive roll stabiliser tanks should be developed, standard methods of predicting roll in irregular waves also be established. The reasonable way for comparison ship roll performance and an ti -r ol li ng effe ct ive ne ss of various type stabilisation system is to be taken the long-term ship roll angle and long-term anti-rolling effectiveness.

by moans of the method which is qlven in Sect ion 2.6 , and to i nte gra te it can alpo been given. In Pig. 6, the ty pi ca l long-term exc ess probability distribution (or return period) of si gn if ic an t ro ll amp litude and expec ted maximum peak value have been given which- is cal cul ate d from Fi g. I TakahashJ.'s f u l l - s c a l e measured data.

11 to

- !•

- 11

/ " \ ^»»-w

1,0

« •

V

1

Evaluation of Ship Rol l Angle In Rules For Stability Of Sea-going Ships

10* 

I0' s

Fig.6

In case the ship principal parameters and conditions have been known, the standard deviation of roll for various ship situation, wave parameter, heading to wave and ship speed e t c . can be cal cul ate d by means of non -li nea r method which is given in Se ct io n 3. Based on wave data of ship shipping route which la given in se ct io n 4. Thus, the long-term exc ess probability distribution of the standard deviation of roll or mean square value of ship rol l amplitude can be gi ve n by meanB of any method in sec ti on 2 and to int eg ra te i t .

 \ \

Sign ifi can t Moll Jwnplltu-3*

» « 0

s.l

« x .  P*«k  v«tu* a  ^ s .

l» ?»

10*'

•«turn P«t»od (or Noll   MUPMT) 10» 10*

10° l0* f »cut probability

10

10"'

\ V s

*%I

long-term prob abil ity dis tri but ion for roll amplitude

One can be take n a sp e c if ic exc ess probability or return period for the long-term probability distribution of the standard de vi at io n, , o, or mean square root of ro ll amplitude, R, thus the corresponding a  or R can be gi ve n. The pr ob ab il it y that . standard deviation, o, of roll will exceed a specified value, Oi, can be calculated from equation (14). Based on diff eren t e xcess probabilit y lev el, Oi, one can be given corresponding o from the long-term probability distribution of the standard de vi at io n. Thus one can be found other

T he long- term ex ce ss p ro ba bi li ty di st ri b ut io n of the expe cted maximum peak value

52 -

reran maximum amplitude of soil according to Por  example,  the Bsyleigb distribution; law. •nes«» probability level la 0.5% mean mairtmni amplitude ot   toll which can te applyed to  Ute requirement» of roll angle in the rule* Cor stability of sea-going snips,* Supposing one applies the expected maximum peak of roll for  long-term excess probability distribution, for   example the excess probability lev el which can Be taken ».5%. Thus th e critical level for ship experienced roll number (or return period) in her lifetime can be given by Beans of e quatio n (13 ) or equ atio n (14) and to calculate, extreme value which can be applied to the requirement» of the roll angle in the rules for stability of sea-going ships. m fa ct, th e to ta l number of r ol l (or r etu rn period), N, experienced by a ship in her lifetime usually lies between lo4  and lo 6 .

exe*»» probaMMiy

Pig.7

Long-term probability distribution of roll amplitude for Ship A and B

the roll damping is function of   roll amplitude. So the calculation for roll damping usually is taken a specific roll damping corresponding to a specific roll amplitude, thus the corresponding roll response amplitude operator can be given

5.2

Assessment of Roll per for mat lon Anti-rolling System Effectiveness

from roll decayed curve.

And

But in case of the ship roll in irregular wave,  the roll amplitude is a variable for the

The ship roll performatlon can be calculated by means of long-term probability distribution for the standard deviation of roll, o, or mean square root of roll amplitude, R, and sig nif ica nt ro ll amplitude $1/3. . If a spe cif ic excess probability, o t , or the total roll number, K, have been known, thus the corresponding o, R or #1/3 can be given from their probability distribution respectively, then make a comparison between the a  (R or «1/3) of different design plan or ship condition. Por ship with anti-rolling system, the ca lcu lat ion for sho rt-term., standard de viatio n, a,"  of rol l have to consider a nt i- ro ll in g characteristics, but it don't have to use non-linear method, if there Is a good anti-rolling effectiveness. The ty pi ca l re su lt i s shown in Pig. 7. Where (* i/ j) A) , («v/3)Bi and (»i/3)A2, («1/3)82 are s ig ni fi ca nt ro ll amplitude for Ship A and Ship B corresp onding to N = jo2  and N « 10 4 . Thé «oA and «oB represents the roll angle extreme va lue for sh ip A and Ship B which i s applicable to stability criteria in rules for stability of sea-going ships. 5.3

Progressively Damping

Approximate

Method

For

Bame sea state, thus the roll damping also is a variable,  it is evident that calculate   roll response amplitude operator by the constant roll ' damping

which

is

not

in

agreement

with

the

actual situation. Therefore, we suggest that the roll damping and corresponding roll amplitude can be .given by progressively

approximate

method

for

ship

in

irregular wave. The first approximate. toll

damping

N 0  =

One can be taken the

f(« 0 )

corresponding

to

a

specific roll amplitude « 0  which is applicable to calculate roll response amplitude operator. Thus, 

the

standard

corresponding given

by

deviation

of

to a specific sea state can be spectral

corresponding

analysis,

significant

roll

and

Roll

Pig.8

53 -

the

amplitude

(«1/3)1. as shown in Pig. e.

In the case of the ca lc ul at io n for short-t erm standard devi ati on, of ro ll , the selection of roll damping is of great importance to ro ll response amplitude opera tor. Because

roll

Decline curves for rolling

The second approxima te. By means of the f ir st approximate ro il damping, Nt » f( (#1 /3) 1), corresponding to si gn if ic an t rol l amplitude (#1/3)1, the roll response amplitude operator and significant roll amplitude, which is in same with present sea state, can be given. The standard deviation of roll corresponding to r o l l damping Nj « f(( #i/ 3>2 ) can be given by means of the ste p by ste p ca lc ul at io n. Usuall y, i t wi ll be su ff ic ed trough three times calculation. we also suggest that the roll damping corresponding to significant roll amplitude can be calculated by progressively approximate method for the roll response amplitude. We make a comparison between r o l l resp onse amplitude operator with non-linear and one with linear which ought to calculate based on non-linear damping NÎ N » f((#i/3) 2 N ) and linear damping NaL  = f((#i/3>z L ) respectively. It is evident that assessment the roll damping effect on roll ought not to use N 0 N  • f(#o) •»*>

REFERENCES

1.

Och l. H.K. s Chang, M. 3. , "Note on th e Statistical Long-term Response Prediction". Int ern atio nal ship buil ding progres s. Vol. 2S, Oct. 1978, No.290, pp. 270-271.

2.

Takoho shl, Y. , "Pul l Sc al e Measurements of à con tai ner Ship". The 2nd In te en l. Sympo. on Pra ct ica l Design In Ship build ing, 1983, Tokyo s Seo ul, Procee ding, pp. 517-524 or J. of Soc. Nav. Arch, of Japan, vol. ISO, Dec. 1981. pp. 327-332 and vol. 152, Jan. 1983,  pp. 268-274.

3.

Roberts J . B . , Dacunha N.M.C. and Hogben., "The Estimation of the tong Term Roll Response of a Ship at sea". NMI Report No. 169, Dec. 1983.

4.

Nbrdenstr'on, N. , "Methods for Pr ed ic ti ng long Term Distributions of Wave loads and Pro bab ili ty of Failu re for ship ". net Norske Veritas, Res. s Dev. Rep. 71-2-s,

6.

vassllopoulos, v., "Ship Rol lin g at Zero Speed i n Random Beam se as wit h Non- li ne ar Damping and Rest ora tion ". Journ al of Shi p resear ch, Dec. 1971, pp. 289-294.

7.

Yamanouchi, Y., "Some Remarks on the Statistical Estimation of Response runctlon of a Ship ". Proc eed ings , Fi ft h symposium on Naval Hydrodynamics. Bergen, Norway, 1964. p.97.

8.

T.K., "Equivalent Linéarisâtion techniques". Journal of the Acoustical society  of America, vol.35. No 11, Nov.

1963,  p. 1706. 9.~

Yamanouchi, Y. , "On th e Ef fe c t of Non-linearity o£ Response on Calculation of the spectrum". Proc eed ing s, l l t h ITTC, Tokyo, Japan, 1966, pp.3897-390.

10. 

Flower, J. Ö. , "A Per tu rba tio n«! Approach to Non-linear Rolling in a stochastic Sea". ISP Vol. 23, 1976, pp. 209-212.

11.

Crand all. S.H., "Perturbation Techniques for Random vi br at io ns of Non -li nea r Systems" . Journ al óf the Aco ust ica l So ci et y of America, v ol . 35. No. 11, .1963, pp.   1700-V705.

12 .

Roberts," J. B. , "A st oc ha st ic Theory for Non-linear ship Rolling in Irregular Seas". JSR v o l . 26, N0.4V, Dec. 1982, pp. 229-245.

13 . 

Haddara, M.R., "A Modif ied Approach for the Application of Fokker-Planck Equation to Non-linear ship Motion in Random waves". ISP Vo l. 21 , NO. 242 , 1974, pp. 28 3-2 88.

14.

caughey, T.K.. "Der ivat ion and Appl icat ion of the Fokker-Planck Equation to Discrete Non-linear Dynamic systems subjected to White Random Excitation", Journal of the Aco ust ica l s oc ie ty of America, v ol . 35, . NO.  11, 1963, pp. 1683-1692.

15. 

Haddara, M.R., "A Note oh th e Power spectrum of Non-linear Rolling Motion". ISP VOl. 30, NO. 342, 1983, pp. 41-44.

16.

vas sll op ou los , t,., "The Application of Statistical Theory of Non-linear Systems to Ship Motion Performance in Random Seas". ISP Vo l. 114 , NO. 150, Feb. 1967, pp. 54-65. .

J971.. 3.

Ochl, M.K., "P ro ba bi li st ic Extreme Values an d t h e i r ! I m p l i c a t i o n for of fs ho re Struc ture "Design", Proc. 10th Offshore '' Techn. confe ren ce, ore 31 61, 19 78. pp; .. .987-989. .

Caughey,

"On son -li ne ar Ship Motion Hasselman, K., in Irr eg ula r Waves". JSR Vol . 10 , No. 1, March 1966, . pp. 64-6B.

Hogben, N. s  Lumb,  F.E.i <<0cean Wave Stat 1st les>> . Natio nal Ph ysi cal laboratory,  H.H. stat ione ry o ff ic e, London, ^~ ' 1967.

27. 

M.K. a n d  Ochi, H u b b l e ,  E . N . ! "six-parameter wave Spectra". Proceedings on  c o a s t a l o f  t h e i s t h C o n f e r e n c e Engineering, Hawaii,  1976.

28.

"An  Analysis of the Morenshlldt, v . A . t Results o f   Model and Full -sca le te st s with Various s  Types Of  S t a b i l i z i n g T a n k s" . 14th  

"Die   E i g e n s c h a f t e n  der walden, H.t Meerswellen i n Nord atla ntis che n Ozean". DeulBcher Wet ter dien st, See wett eran t, Einzervoeroffentllchungen Nr. 41 ,  Hamburg (1964).

29 . 

Fang, Z . S . S Hogben.t "Anal ysis   and o f   Long Prediction Term Probability Distributions of   wave Heights and  Periods". NMI R146,   Oct. 1982.

Haver, S. i "Wave Climate o f f   Northern Nor way. Applied ocean Research,   1985, v o l . 7 No. 2. Yamanouchl. Y. 6 Ogava, A.  i  "statistical Diagrams on the   winds and wave  on the  North Pacific Ocean "  Papers o f   Ship Research I n s t i t u t e , No. 2»  1970, Japan. SRl63i <
Kawabe, H. ,  Mano, H.  arid  Awa, K. i "On th e Variety o f   Wave Conditions Encountered by Ships sailing in the  Same Se a  Zone". J. of Soc;  Nav.  Arch, o f   Japan, v o l . 152, Dec. 1982. '' Ochi,  M.K.: "Pr obab ilis tic Extreme Values f o r  O f f s h o r e an d t h e i r Implic atio n Stru ctur e Design". Proc. lot h Offshore Techn. Conference, OTC 3161,  1978.

- 55 -

ITTC P r o c e e d i n g s , V o l . 4  OTTAWA,  1975.

" Se a  t r i a l s on th e Williams. I. K. i Fi sh e ri es Research Ve ss el "Core11a". BSRA, Report NS.363 o r Naval Archi tec tur e Report No.  93, 1972.

Third International Conference on Stability of Ships and Ocean Vehicles, GdarisJk, Sept. 1986

bj'fc O) IO)\o

~i"aper 1.9 THREE.DIMENSIONAL NUMERICAL SIMULATION OF GREEN WATER ON DECK

J.T. Dillingham, J.M. Falzarano

ABSTRACT A method i s de sc ri bed to model the motion of a sh ip with water on deck . Pr ev io us ly , the method of Glimm was app lie d by the f i r s t author to so lv e the problem of water sl os hi ng on the deck of a sma ll fis hin g ve sse l in order to study the ef fe ct of the deck water on thé ve ss el 's st ab il it y and motion in wave s. Glimm's method i s a numeric al scheme for solving the hyperbolic equations associated with the shallo w water flow and i s noted for i t s a bi l i t y to handle complica ted jump phenomenon e f f i c i e n t l y . In th is paper cont inuing work is desc ribe d which exte nds Glimm's method from the r e st r ic ti ve two dimensio nal problem sol ved pr ev io usl y to the more gen era l t hree dimensional ca se . The method has appl ica tio n to the pred icti on of the ef fe ct of deck water on various type of ve ss el s, es pe ci al ly those with larg e f la t deck are asThe re su lt s of 'three dimensio nal flow sim ula tio ns are prese nted for se le ct ed c as e s . In order to make Glimm's method useable for eval uatin g the saf et y of small ve ss el s; the deck water sim ula ti on has been combined wit h a ship motion sim ulat ion. Limited res ul ts are prese nted along wit h a de sc ri pt io n of the program and its suggested application. 1 . 0 

BACKGROUND

-,

The g re at es t pe ri l to ships at sea is inadequate st ab il it y and the pos si bi li ty of cap siz ing . Toda y, most s hips which are designed according to national and inte rnati onal sta bi li ty standards are sa fe for the most par t. However, there are ships, especially the smaller ships, which do ca psi ze . The safe ty si tu at io n for ship s has improved sin ce the c l as s i c a l work of.Rahol a in 1939 1201 ye t the methods of ana ly si s have changed l i t t l e since Moseley [12] (1850). Typic ally ship dynamic st ab il it y ana lys is is s t i l l * based on the assumption that a l l ups et tin g moments are applie d st a t i c a l l y . To account for dynamic, ef fe ct s fact ors cf sa fe ty are applie d. These cor rec tio n fact ors are arrived at by examination of suc ce ssf ul ships and ship cas ual ti es

exp er ien ce) . Yet cor rec tion s upon corr ecti ons are applied so that one quickly los es sig ht of the physics of the phenomenon which is being modeled. An al te rn at ive approach to evalu ate ve sse l st ab il it y is the Use of computer simulati on to pre dic t v es se l motions under various co ndi tio ns. Unfort unately the majori ty of computer simu lat ions are based on small amplitud e.moti ons of the ve ss el as we ll as smal l amplitude wave theo ry, and are the ref ore li ke ly to be in se ri ou s error when the motions are large enough to endanger the ship. Early re sea rch er s found the need to use time domain shi p motion programs t o a cc ur at el y model non -li nea r motion leading to ca ps iz in g. One of the projects investigated the so-called synchcronous r ol li ng phenomena expe rie nce d by shi ps in a fo ll ow in g se a. Between 1970 and 1979 the U.S. Coast Guard spons ored an ex te ns iv e amount of advanced st ab il it y research at the University of Calif ornia , Berkeley under the direction of   Prof.  J.R. Pau Hi ng . Along with numerous experime ntal r es ul ts and te ch nic al rep orts th is resea rch produced the computer program CAPSIZE. CAPSIZE i s a time-domain shi p-m oti on computer program which models the phenomenon of autoparametric e xc it at io n. . This phenomenon may have caused the lo ss of at le as t one general cargo ship (the S.S. Poet) [13]. CAPSIZE i s capable of sim ula ti ng var ious t ypes of nonli nea ri tie s, part icul arly the nonlinear changes in hyd ros tat ic rig ht in g moment which res ul t from l arg e amplit ude' r o l l motions and from the change in .w ate rpla ne shape during the passa ge of a wave c r es t . Linear frequency-domain sh ip motion; programs must assume const ant c oe ff ic ie nt s, and consequently cannot account for such phenomenon. The phenomenon of autopa ramet ric ex c it at io n and the computer program us ed .t o model i t have been mentioned for two reaso ns. Fi rs t, the exi ste nce of autoparametric ex ci ta ti on reminds Naval Arch ite cts of the comp lexi ty of modeling the extreme motions leading to ship cap siz ing. Also, time-domain si mu la ti on programs like CAPSIZE are the onl y methods by which one can hope to ac cu ra te ly model (i.e.

- 57 -

mul tip le, , complicated nW- li nea r phenomena leading to capsize events. The development of th e program CAPSIZE and the associated research represent the most extensive inv est iga tio n of it s time. Since that time the Uni ted Kingdom and Norway have undert aken ot he r ex te nsi ve research programs int o the causes of ship c ap si zi ng . The Unite d Kingdom's Board of. Trade has sponsored numerous th eo re ti ca l, applied and experimental investigations for over ten years under th e t i t l e of SAPESHIP. The SAPESHIP pr oj ec t i s desc ribe d in [21] and by Odabassi [1 8] , includ ing a summary flowc hart of the in ve st ig at io n plan and an an al ys is of the numerous ca ps iz in gs which prompted the res ear ch. The sp ec if ic cas ual ty that prompted the pr oj ec t i s the ca ps iz e of the M/T Edith Terkol which apparently motor exceeded all applicable s t ab il it y re gu la ti on s. The emphasis of most work has been on the appli cati on of cl as si ca l rigid-body dynamic stability models to ship stability analysis. The Norwegian research tras prompted by the capsizing of the M/S Helland-Hensen, which also apparently met a l l a ppl ic ab le Norwegian and in te rn at io na l (IMOj Torremoli nos) stand ards [7 1. The Norwegian "Ships in Rough Seas Pro jec t" has focused i t s emphasis on breaking waves and modeling r e a l i s t i c extreme sea con dit ions leading to ca psi zin g. This work is summarized in (221. This. pape r de sc ri be s ongoing work t o improve methods for sim ula tio n of gre en water on the decks of ve ss el s at se a. The motivat ion for th is work began with the desire to explain the numerous capsi zing s of small fis hing ve ss el s (see Storch [25 ]) and to ge ne ra ll y improve the understanding of ' the st ab il it y of those types of ve ss el s. However, i t has broader app lic at ion s sin ce the method descr ibed can in pr inc ipl e be used to numerical ly dete rmin e the beha vior of water on the deck of any vessel. In some ca se s wate r on deck may have; a significant effect on safety or operations. The so lut io n of the water-on-deck problem wi ll by no means comple tely sol ve the ship saf et y problem, si nc e ca su al !t es usu al ly happen because off . a number of extreme si tu at io ns which occur simul tane ously . Some ty pi ca l documented cap siz ing inci dent s seem to substant iate t his c laim. PATTI-B [14] ca psi ze d whi le anchored cl os e to shore.. At le ast -t hre e ef fe ct s are believe d to have ca us ed .t he in c id en t . The PATTI-B was anchored by the st er n, li mi ti ng i t s freedom, and anchored in shall ow water where waves ste epe n due to sho al ing . A Coast Guard l i f e - b o a t was sta ndi ng by when the ve ss el caps ize d. The event s leadin g to the cap size were as fol low s: the ve ss el took water over the st er n, which was momentarily trapped between the bulwalks enc los ing the lar ge open af te r deckt the ve sse l became poised on a wave, lo st st ab il it y, down-flooded and capsized. All of these events

occurred in a very short time in te rv al , sugges ting th at dynamics were imp ort ant . Furthermore, the PATTI-B met or exceeded a l l ex is ti ng c ri te r ia in cl ud in g the Coas t Guard weathe r c r i t e r i a and IMO Torremolinos. The JOAN-LA-RIE II I a l s o ca ps iz ed bec ause of wate r on deck and oth er e f f e c t s . The JOAN-LA-RIE II I was fi sh in g cl os e to a number of othe r ve ss el s ; when she was h i t by a brea king wave, hee le d ov er , accumulated wate r in t o her co ck pi t, swamped and sank I In gene ral , ships that cap siz e do so witho ut: [151. numerous othe r ve ss el s looking on. Because of thi s ' many ve ss el s that cap siz e do so without a tr ace . Two such in ci de nt s where the U.S. Natio nal > Transp ortat ion Saf ety Board (NTSB) be li ev es , ve ss el s : prob ably sunk due to wate r on deck are the sin kin g ' of the M/V HOLOHOLO [16] and the F/V AMAZING GRACE [17]. In t he AMAZING GRACE incident the NTSB believes that the most reasonable explanation of its loss is that water bacame trapped on deck due to the st er n ramp door 'and fr ee ing port s being cl os ed . A wave hit the vessel and the vessel,capsized. A fi na l inc id en t where water on deck was a major con tr ibu tin g fa ct or was the lo ss of the M/S Heiland Hansen [7 ] . The ve ss el was hi t by a breaking wave which momentarily poised the vessel at a 60' hee l ang le. Hater f i ll e d the open deck area reducing the ve ss el s s ta bi li t y so that when the ve ss el was hit by another wave i t caps ize d. Other documented l os se s of U. S. , U.K., Norwegian 'and oth er ve ss el s caps iz ing due to water on deck are too numerous to mention; yet all seem to involve small vessels with large open decks enclosed by bulwa rks.. A typ ica l scenari o in a capsizing in ci de nt in vo lve s the occurre nce of two or more st ee p waves in a sho rt per iod of time . I t may be , for example, that the deck water which is present as a r es ul t of the f i r s t wave reduces the ve ss el 's st ab il it y su ff ic ie nt ly that the second wave, may caus e a ca ps iz e. The time con sta nts for the water leaving the deck, the phasing of the vessel's rolling motion, the motion of the deck water and the sea surfac e el ev ati on are a l l important. Since the system is nonlinear, neither a static analysis nor a li ne ar frequency domain an al ys is can determine th is crucial phase information. The phasi ng of the motion of the deck water wit h re spe ct to th e, shi p may be such that i t will. ' act as an anti-roll tank, which for moderate motions w i l l ac tu al ly reduce the amplitude of ro l l. This was shown by the previous work of Dillingham   [9,10]. However, ther e i s stro ng evide nce based on records of actu al cap siz ings that under cer tai n circumstance s the reduction of st ab il it y caused by the deck water, occuring co inc ide nta lly with large • over turni ng moments,, may con tri but e to l ike lih oo d of ' capsize. Yamakoshi, e t . a. [21 ] have conducted model

experiments

to ascertain the effect of

shipping of

water on the stability and likelihood of ' small

fishing

experiments

vessels.

and

Based

the • results

capsize of

oh

of

their

other

own

Japanese

research groups they conclude that shipping of water on deck plays an important role in the

capsizing of

fishing vessels. 2.0

PROGRESS IM THE SOLUTION OF THE DECK  WATER PROBLEM

Up until a few years ago analytical the effects

of deck

 pseudo-st atic in

water were mainly

approximations due to

accurately simulating the

 behavior

of the

studies of

flow.

confined to

the difficulty

complicated nonlinear

Because of

difficulty and

inaccuracy many investigators have opted experiments.

Recently

accurate numerical

Dillingham

time domain

for costly

[9] applied

method

an

for solving

the shallow water wave equations which

describe the

Fig. 1 ,

deck water motion. The so-called  means

to

solve

Two dimensional sloshing

Glimm's

method

hyperbolic

systems

provides of

problem definition

a

equations

approximately by a discretization of the flow region into finite developed By this

cells .

In

[10]

Glimm's

as a two-dimensional we mean

two-dimensional.

Mathematically

one-dimension al

since

space. variables one,

namely

simplification

it

is

the

transverse

is justified

actually

shallow

number, of

describin g

the

is geometrically

the the

was

computational tool.

that the problem

approximation ' reduces

method

water

motion

coordina te.

to

flow

of

athwart direction .

the

deck water

the shallow

was

in

the

The problem was then formulated

as a nonlinear hyperbolic system of water wave theory.

equations using

In two

dimensions

the problem is visualized in Figure 1.

independent

wave

when

. 'predominant

If the deck water is shallow then the following equations result

from satisfying the

conditions of

This

conservation of mass and momentum, and the kinematic

considering only

 boundary conditions on the bottom (see Stoker   [24]):

the transverse motions of the vessel (roll, sway and 3u . 3u 3X  — + u-- + a , — = f_ 3t 3x 3x 

heave).

For fishing

vessels which

are

assymetric for

and aft, their motion may not be well a two-dimensional model.

predicated by

We have, therefore, chosen

to expand our simulation to

six degrees-of-freedom,

3X 3X 3u  — + u — + X — = 0 3t 3x 3x where u

although the roll motion is of principle concern.

2.1

•» the horizontal velocity  point

The problem of computing the flow of water in a  been

studied very exte nsively.

typical

of almost full ballast tanks,

integral technique even

of Faltinsen [27]

for ... large motions

nonlinear

free

., exactly.

the

For large

of the

surface

condition

has not

X

depths,

f x = the

is

shallow water

formed.

case where

This specific

Dillingham

[10] .

In

hydraulic

the

shallow water sloshing back and

dynamic

he .considered

the

forth on

the aft deck as the vessel r olls. '•'.'•'.

It

by gravity as the

t

= time variable

The main difficulty associated with finding the solution to

such equations

hydraulic jumps

is the handling

which inevitably appear.

of the Prior to

numerical ly using the method of characte ristics.

of

effect of

considered.

fluid

force

the appearance of jumps the solution may be computed

study

fishing: vessels

a

the

body

a

of

[10] only

on

(transverse)

problem was investigated by

stability

In

and roll and

deck rolls

satisfied jumps are

horizontal

' exerted

the

However the technique is not suitable for

heave

= the local water depth on deck

is applicable since

resulting from

gravity

the boundary

fluid,

of a water column

a z = the vertical acceleration of the deck at a

Two-Dimensional Problem

confined area such as on the deck of a boat

(1)

constrained to:. heave,'sway

and roll

motions

 problem can

of the

vessel are

be linearized and

small

enough

problem

was

the vessel

was

and

that the

 motions of the depth

59 - :

the deck

in the vertical

vertical direction of

the

deck

are small

wat er.

the

solved analytically.

Unfortunately smallness in this case means

two-dimensional

wa s. assumed, that

the

If

The

that the

direction are

compared linear

to the

theory is

therefore

when a part of the deck

useless

becomes

completely dry. In  [10] a  relatively

EXP  = 107 

new  numerical technique

was applied  to solve these equations.

Tint = 6.0

EXP = l o i 

Tine  ROLL  = 4 . 3 3

ROLL = - 0 . 0 0

*^ji

This method,

known either as the  random choice method  or Glimm'a  meth od, was

 by  Glimm    [11] as  part first introduced 

of a   proof that solutions exist for such

It was   later developed

equations. numerical

into a  useful

by   Chorin  [2,3]. 

tool

systems of

Collela  [4]

investigated methods for  optimizing the  accuracy and rate of  convergence of details of this

the numerical schem e.  The

EXP s 107  TlltE  = 7.0 ROLL  = 4.33

EXP = 107  III« = 7.5 ROLL : 0.00

TinE = B.O EXP s 107  ROLL a -4.33

TIME = 6.S EXP s 107  ROLL = -4.33

method   may  also be found  in Conçus

and Proskurowski [5], Sod    [23], and   Wigton the references cited therein.

[26] and

Ne  mention here only

some of the significant features. Glimm's

proceeds as

method

a  time

stepping

Scheme in   which the state v ariables, the  depth and the

velocity in   each

horizontal water

determined  at  each their

in

values

step  on

time

the previous

of Glimm'3

attractiveness arbitrary

number,

requiring any

it

respect

step.  The

time

the  fact jumps of

hydraulic

  and 

size

of

location

without

special tracking algorithm.

In this

is vastly

characteristics

superior to

which

characteristic the

the   basis

method lies in

it  automatically treats

that

cell,  are

breaks

lines

of

formation

jump.

when  the

down

corresponding to

converge,

a

the  method  of

 numerical errors and 

unconditionally stable

Fi g. 2

method  is

Glimm's

Comparison between

numerical predi ction s (so lid

can be

and experimental res ul ts

lin e) [1]

 and  quantified   reduced  to any arbitrar ily small size  by utilizing time

smaller cell

At  each

steps.

  smaller and 

dimens ions

the solution

time step

advanced  by randomly sampling a  series solutions

the flows in  pairs of  adjacent calls.

of depth

of explicit

the  interactions between

which describe

estimates

is

The resulting

and   velocity, which

are random

comparing the

numerical pr ed ic ti on s

ve loc it y

an

to

equations which

anal yti cal

of

depth

solu tion

is known for a sp ec if ic

The t e s t case i s th e Riemann problem, al so

of (1) as the  number of  time steps becomes large.

the

Caglayan [1] performed experiments to  test the accuracy  of

the  

numerical

hé  subjected   a

experiments

In  these

results.

two-dimensional shallow

tank of  water to   oscillations in roll.

was

depth

measured

photographically

The water at  several

in  time and  compared with . the. results of

 point s

numerical calcula tions. in Figure 2.

Typical results

are shown

indicated that the

These experime nts

 for  short accuracy of the  predictions was quite good  There was some evid ence that over a

time duration s. long

time

span

the numerica l

conserve mass precisely. the

case of

the fishing

did not

solution

This was vessel

hot critical in

motion simulation

since the amount of deck water varied rapidly due to flow Small

over the  bulwarks and    through

the scuppers.:

variations- in mass should  not  affect  thé

overall behavior. • If the

we presume that the equations

physical

examine

processes • accurately

the  accuracy

of 

Glimm's

(1) describe then

we

can '

method  by

— 60

break ing

Figu re 3. on

th e

problem , which

is

these

te st c ase .

over short time spans converge to the exact solution

dam

of

known as

d ep ict ed

We assume th at the wat er depth i s two s i d e s

of a . dam.

inst anta neou sly at

time,

t

and

in

known

The dam . i s removed « 0 ,

and

a solu tio n

des cri bin g the re su lti ng flow is sought.

The exact

an al yt ic al so lu ti on to th is problem may be refer ence s

[8]

and

[ 9] .

found in

.

Using Glimm's method the Riemann problem can be solve d numer ically.

Values of

depth and

ve loc it y

are computed at grid points which are equally spaced along the

x -a xi a.

Figure

3

shows the

depth obtain ed : an aly ti cal ly

water for one

in sta nt in

time .

In th is

val ues of

and numerically te at case

water depths are taken to be 4 f t . and 2 f t . l e f t and rig ht gri d fo ot .

spacing in

on the

sid es of the dam re spe ct ive ly. the hor izo nta l

The numerical

d ire cti on

the

is

The one

re sul ts, are quit e accurate in

sp it e of the appare ntly coar se mesh.

The time ste p

was taken to be .01 seconds.

3 . 2 .  Three-Dimenaional Problem In three dimensions the shallow water equations

TWO DIMENSIONAL RIEMAN PROBLEM

3u 3u 3X  — +'u— = - az, — + f„ 3t 3x 3x

NO VISCOSITY

 \  \V

».» •

sa-

K\

J.J3.83.»3.«3.3J i  l l . 3-

3X 3X 3u — + u— = -A-3t 3x 3x

THEORY M 

V

NUMERICAL

 \ \  \ \  \

which express  momentum

2.82.7 2.6 •

y

the requirements

and mass

3v 3u  — + v — 3t 3y

of

conservation of

in the x-dire ction.

During the

3X -a z --  • fy 3y

*-i-»-r •»

(4a)

3X 3X 3v  — + v — - - X — 3t 3y 3y

DISTANCE FROM DAM (FEET)

Fig. 3

(3b)

time step the following equations are solved:

as2.4-

2.3.2.22.1 2 -

(3a)

which correspond

Comparison between analytical

and numerical solutions of two dimensional Riemann problem

(4b)

to conservation

of

momentum

and

 mass in the y-dir ection. —Each time step consists of an

x-sweep across

followed by à i  value).

take the foll owing form ( refe r to the de fi ni ti on sketch in Figure 4)t 3u 3u 3u 3X — + v— + u— (2a) + 3t 3y 3x *« 3v 3v 3v 3X — + v ~ + u— = - a _ ~ + f v (2b) * 3t 3y 3x 3y 3X 3X 3X 3v 3u — + v— + u— + X— + X— = 0 (2c) 3t 3y 3x 3y 3x where v » the longitudin al v elo cit y of a fluid column f„ ». the hor izo ntal lon gitu din al body force exerted on the fluid by gravity

"Sx

velocity

each row

y

(constant

During

the

value)

sweep across each column (constant the x-sweeps the y-component of

is transported as a passive

during

j

y

sweeps

the

scalar, while

x

velocity

is

transported as a passive scalar.  A

fairly

demanding

test

of

the

three-

version of'Glumm's method  :a the three-

dimensional

dimensional Riemann problem with the dam oriented at an angle to the mesh. the

same, except

 perpendic ular depths are

Analytica lly the solution is

that the

to the

velocities

are vectors

initial dam posit ion

and the

constant along any line parallel

to the

initial dam position Numerically,

this represents a

fairly general

case, since there will be interaction  propagating in the

x

and

y

between waves

directions which must

ultimately" average out. to produce

a

well defined

disturbance .prorogating in a direction perpendicular to the dam. for

Figure 5

this test

shows the

case compared

numerical results

with

the theoretical

results (which are the same as in Figure 3 ) . In Figure 5 the depth is shown as a function of distance from

Fig. 4 Three dimensional

THREE DIMENSI ONAL RlEMAN PROBLEM , •

sloshing problem definition

To extend the

method

suggested by

Glumm's method

of

operator

separate

three dimensions

splitting

Collela [41. Operator

for : separating two

to

problems

longitudinal

flow.  : The

illustrati ve

purposes is

for,

the

deck taken to

is

used

as

'.'if

splitting calls,

the multidimensi onal

the dam along a line perpendicular to

problem

2.B 2.7 2.« -\ 2.52.4 • 2.3 2.2 2.1 -

into

transverse and

area,

which

for

be

square,

is

39 38 • 3.7 3.« 3.53* • 3-3 32 3.1 3 -

:

^>\ V «

M .

- \ S

K

at

A single

consists of first, a step of duration y-direction.

'During

the

,

v

_>:-.-!V

'A

0.0

0.1 0.01 0.001

* 8

>

 \  \*

'

0 DISTANCE FROM 0AM (FEET)

time step of duration At

x-direction followed- by a step of duration the

* • *

. *

divided into a rectangular mesh in two dimensions as shown in Figure 4.

-WITH VISCOSITY

x.

time

Fig. 5

in the At

step

Comparison between

analytical and numerical solutions

in

to three dimensional Riemann problem

the

following equations are solved:

-61 -

x

»

the dam. The

the

operator

sp li tt in g technique is quite suc ces sfu l as

a method

to

re su lt s

extend the

ind icat e

c ap ab il it ie s of

three- dimensi ons although good

as

that

for

the

dis cr ep an cie s

are

Glimm'8

the accuracy

two -dim ensi onal seen

not se ver e

and

appear to

expected .

This is

is

not

problem .

between

numerical so lu ti on s in Figure 5.

method

the

to as

- Some

exac t

and

However the se are

be random

in

natu re as

hot en tir el y undesireable since

th e rea l flow w i l l be tu rb ule nt and hence

random in

the vicinity of the jump. 6  shows

Figure snapshots  of

a 

the  flow

at intervals  of

one   half

Qualitatively the  reflection of the  shock

second.

 the   side walls from 

waves

of  numerical

sequence

seems

correct although

analytical results are not  available for comparison. figure also shows the interaction

This

one being

shock waves,

the reflect ion

 bein g

between two

the original shock from  the  side

and one

wall.

of numerical viscosity

has been

introduced into the calculation procedure .

This is

 A

small amount

a

essentially

rather

simple

spatial

smoothing

It was  suggested   by  Collela [4] that this

function.

would improve

the numerical

observed that

the principle effect was to  make the

graphical representation

accuracy,

in Figure 6

however, we

more pleasing

to look at.   Figure 5 shows a  comparison of results values of the   nondimensional viscosity

for various  para mete r. 3.0

Application

 Now

we   have

that

 behaviou r  of relatively

the   deck

a method  to  compute  the water

straightforward

it is

numerically

<-fco inco rpor ate

this

 procedu re into any time-domain 'ship-motion program. The forces exerted  by the deck water on

the  vessel

can be easily computed  and the  effect on

the  vessel

 motions thereby de term ined.

The  motions of the deck

water are in  turn affected  by the vessel moti on. simple procedure volume of

for approximating

the deck

the  change

water resulting from

A in

flow over

the bulwarks and   through the scuppers is  decribed   in [10]..

 At

the University

of Michigan

we   have

just

 begun  a project in  which we will try. to . utilize the numerical methods described above for simulating the  behavio ur of  water understanding vessels.

of

on deck in order to  improve our stability

This project

U.S. Coast Guard.

criteria

is being

The approach

for  small

sponsored    by the

 to which we intend 

take is summarized  by the following s teps: 1) Identify well documented incidents of.vessel capsizing, is

especially those where water on the deck

suspected  to  have 2)

Utilizing

been

part . of

numerical.

the  cause.

simulation

try to

recreate the incidents. 3)

Identify

the mechanism  or  primary forces

which resulted in the capsizing as determined by the

would also

simulation* ,

Guard for their support.

4) By

parameter variations identify changes in

Determine

the

applicability

of

Dr.

existing

Identify reasonable changes in the stability

and

the

is necessary to include as

real nonlinear!ties in the equations

 motion as

possible.

available

we

will

formula tions.

At

degrej-of-freedo m the

Where exact use the

ship motion

impulse response

Perex [12]) is various

simulati on.

The

advantage

of

for

impulse

being

not

a

simulation

the

a

water

domain

flow

dependence

and

added

mass

coefficients of the ship hull. a

variables can  might

arise

function of

3.

of

easily be incorporated. from

characteristics ,

a mooring

line

nonlinear

the state

Collela, Phillip, "An Analysis of the Effect of Operator Splitting and of the Sampline Procedure on the Accuracy of Glimm's Method," Ph.D. Dissertation, Department of Mathematics, University of California, Berkeley, 1979.

5.

Conçus, Paul and Proskurowsk i, Wlodzimeierz, "Numerical Solution of à Nonlinear Hyperbolic Equation by a Random Choice Method," LBL-6487 Rev., Lawrence Berkeley Laboratories, Berkeley, California, December 1977.

6.

Cummins, W.E ., "The Impulse Response Function and Ship Motion," Schiffstechnik, Band 9, Heft 47, June 1962.

Such forces

with

nonlinear

hydrostatic

restoring

forces, wind heeling moments, etc..

 At a later attach the flow

point in this project w e

subroutine GLIMM,

of

the deck

 program.

CAPSIZE

simulation effects

of

which

water, to will

some

the

the

original CAPSIZE

provide

of

resulting from

the

intend to

simulates

a

more

important

. 7.

Dahle, E.A. and Kjaerland, 0., "Capsizing of the M/S HELLAND HANSEN," RINA Transactions, 1979...

8.

Dillingham, J.T. and Falzarano , J.M., "A  Numerical Method for Simulating Sloshing," to  be published: SNAME Spring Meeti ng, 1986.

accurate nonlinear

the time dependence

of the

shape of the displaced volume.of the ship. •'.' failure

capsizing in result of

of

stability

criteria

certain instances

to  • prevent'!

is likely

either oversimplification

Chorin, Alexandre J., "Random Choice Methods with Applications to Reacting Gas Flow," Journal of Computational Physics, Vol. 25,  No.  3, November 1977, pp. 253-272.

4.

damping

Since the simulation time or

is a graduate

Chorin, Alexandre J., "Random Choice Solution of Hyperbolic Systems," Journal of Computational Physics, Vol. 22, No. 4, December 1976, pp. 517-533.

is in the time domain any type of force which can be described as

of

2.

simulation the frequency

the

M. Falzarano

University

Cayanhan, I., "Effect of Water on Deck on the  Motions and Stability of Small S hips," Doctoral Dissertation, University of Washington, 1983.

has the

technique which accounts properly for of

The

• 1.

test the

response method time

Mr. Jeffrey

at

Naval Architecture

six

based on

and

deck

Engineering

of

an assistant

REFERENCES

empirical

time

to refine

Dillingham is

student in the same department.

method (Cummins [6], Perez y

being used

algorithm s

methods are

present

United 'States Coast

many of of vessel

reasonable

T.

the Department

Marine

 Michi gan.

criteria which might have prevented the accidents. He feel it

Jeffrey

 professor in

stability criteria to the given situation and vessel type.

thank the

 ABOUT THE AUTHORS

the design which mig ht have prevented the capsize« 5)

like to

(i.e.

to

,9.

be a

use of

Dillingham, J.T., "Motion Prediction for a  Vessel with Shallow Wate r on Deck," Ph.D. Thesis, Department of Naval Architecture, University of California, Berkeley, 1979.

simple empirical formulae for stability criteria) or overgeneralization to different  methods

(i.e. applying the

types of vessels).

which

we are

same formula

Dillingham, J.T., "Motion Studies of a Vessel with Water oh Deck," SNAME Marine Technology, February 1981.

11.

Glimm, J., "Solutions in the Large for  Nonlinear Hyperbolic Syst ems of Equa tions," Communications on Pure and Applied Mathematics,  Vol . 18, 1965.

12.

Mosely, C., "On Dynamic Stability and the Oscillation of Floating Bodies," Philosophical Transactions of the Royal Society, London, 1850.

13.

National Transportation Safety Board Report, "Disappearance of the U.S. Freighter SS Poet in  North Atlant ic'Ocean, Abo ut October 25, 1980."

Unfortunately the

suggesting to

use

are

not

accessible to naval architects or operators as rules of thumb or simple formulae.

10.

Such simple

formulae

are useless, however if they do not reflect physical reality

over the

range of

circumstances

to which

they are applied.

It may be that the most reliable

stability criteria

will ultimately

requiring

that each

vessel design

be

achieved by

pass

through a

"standard" computer simulation without capsizing.  ACKNOWLEDGEMENTS The

authors would

Foundation supported The

and a

the

portion

University

Consortium which of

r 14. like to thank

various of

the Seagrant

companies

this

of . Michigan's

research

who

have

through

Seagrant/Industry

is administered by. the Department

Naval Architecture ^ nd Marine Engineerin g.

We

63

National Transportation Safety Board Marine  Accident Re port, "Grounding and Capsizing of. the Clam Dredge PATTI-B, Ocean City Inlet, Ocean City, Maryland, May 9, 1978.V  NTSB-MAR-79-9. .

15.

National Transportation Safety Board Marine .  Accident Report, "Sinking of the Charter ;' Fishing Vessel JOAN LA RIE III, off Manasquan Inlet, New Jersey, on October 24,  1982,"  NTSB-M AR-84/02.

16.

National Transportatio n Safety Board Marine  Accident Report, "Sinking of the M/V HOLOHOLO in the Pacific Ocean near the Hawaiian Islands, December 1978," NTSB-MAR-80-15.

17.

National Transportatio n Safety Board Marine  Accident Report, "Loss of the U.S . Fishing  Vessel AMAZING GRACE abo ut 80 miles east of Cape Henelopen, Delaware about November 14, 1984," NTSB/MAR-85-07.

18.

Odabassl, A.Y., "Ultimate Stability of Ships," RINA, Transactions, 1976.

19.

Pèrez y Perez, Leonardo, "A Time Domain Solution to The Motions of a Steered Ship in  Waves ," U.S . Coast Guard Report N o. CGD- 19-73,  November 1972.

20.

Rahola, J., "The Judging of the Stability of Ships and Determination of the Minimum Amount of Stability," Doctoral Thesis, Technical University of Finland, 1939.

21.

Safeship Seminar Proceedings, National Maritime Institute, Feltham, March 4, 1982.

22.

"Ships in Rough Seasi Project Proceedin gs," RINA Occasional Publication #5, February 5, 1982.

23.

24.

Sod, Gary A., "A Numerica l Study of a Converging Cylindrical Shock," Journal of Fluid  Mechanics, Vol . 83, Part 4, 1977, pp. 78 5-794. Stoker, J.J., Water Waves, Óuterscrend inc.. New York, 1957.

: Publishers,

Storch, R.L., "Alaskan King Crab Boat Casualties," Marine Technology, SNAME, January • - '• -1978. •...••' 25.

26. 27.

U.S . Coast Guard, "Marine Casualty Reports JOAN LA RIE II I," Washington, DC, 1984.

M/V

Wigton, Larry, "Glimm's Method for Humans," lecture notes prepared for course on numerical  methods. Depar tment of Mechanica l Engineering; University of California, Berkeley, California, 1979.

Faltinsen, Odd M., "A Numerical Nonlinea r  Method of Sloshing in Tanks with Two. Dimensional Flow," Journal of Ship Research,  Vol.- 22, No . 3, September 1978,, pp. 193 -202.

28.

- 64 -

Third International Conference on Stability of Ships and Ocean  Vehicles, Gdatisk, Sept. 1986 Paper 1.10

APPLYING LYAPUNOV METHODS TO  INVESTIGATE ROLL STABILITY S.R. Phillips

'ABSTRACT

where

This paper describes how a roll bound may be ' obtained for any ship, without recourse to Iter ative techniques, it concludes that a large scale statistical analysis of ships will produce a realistic and usable stability criterion, based • upon Lyapunov theory. 1

INTRODUCTION

This paper will consider, In practical terms, the problem of existence of à Lyapunov.bound for the roll motion of a vessel in beam seas, obeying the simple second order equation 0 + f(9) 9 + g(9) - e(t)

.•x  - /h'(F-h)  (2/| 8 |  - /hHFhT) d.2 - /h'(P-h) (2/|g| + /h'(F-hi)

(3)

He can eliminate the dependence of the excitation on time by considering that e(t) always assumes Its maximum absolute value. Call this value the "Excitation Lever". Plots of 4>j, <|>2 and the excitation lever are shown In Fig.l.

(1)

where f(9) - the linearised damping function g(9) - the restoring function e(t) - the excitation function

HAX

Our Lyapunov function will take the formt . V(9,S) - j [9 + P(9) - h(9)]2 + 0(9)

(2)

where

 MAX: 9

"

•

.

'

-

.

•

'

F(9) - ƒ f(x) dx .• -• •: '.

x-o

e 0(9) - ƒ g(x) dx x-0 h(B) » a9 + b93 where a,b are arbitrary constants to be discussed in more detail later. 2

Figure 1

Characteristics of Lyapunov Diagram

FORMULATION OP THE LYAPUNOV BOUND Equation (3) is a necessary but not a sufficient condition for stability. Reference [l] 'goes on to discuss other necessary conditions, but these are all dependent on the distance between 8j and 92  being reasonably large. This distance, which decreases with Increased excitation, we will call the "Stability Range", and Its sire may be considered as an indication of the likelihood of stability.

Reference [l] proves that for' a Lyapunov ' bound to exist It Is necessary that there be a region of 9 in which -h*(F-h) - 2/gh'(F-h) < e(t) < -h*(F-h) + +2/gh'(F-h) This bound may be expressed as:

-(J,2  C e( t) < $ x -((.j < e ( t ) < 4,2

 As the Stability Range is dependent on the j curve alone, .. and the Lyapunov diagram is symmetrical, then It is usual to confine our study to the positive quadrant alone.

if g > 0 if g < 0

- 65

3

VALIDATION

0.80

Because the h  fonction le  arbitrary, there • are. an  infinite nimber of $  curves that may be  plotted. Our aim   wist be to choose an h function «bleb,  tor-jea 4>j to be  large and    therefore «axl•lses the   likelihood  of  obtaining a  large stability range.

 METKE9

 At  BMT,  software has been developed that employs the   above theory to  calculate  the stability range for a number of values of a and b (the h  coefficients), and    Iterates In a  fashion tending to maximize this range. A   Lyapunov bound  may then be  fitted    If the range of stability Is large enough.

110.0 DEB

This criterion can be relied upon to produce a realistic roll-bound  for any given vessel. Time simulations of the  equation of motion (1)  assuming a triangular wave profile, repeatedly produce a   ; roll motion exceeding 90Z of the Lyapunov bound calculated. Model tests are  planned  for  more rigorous verification of the  method.

Figure 3  Ship No.5 - Low  Damping

 As a  simple demonstration of the   robustness 'of the method, take Example Ship No.5 (particulars | In Table 2,  section 4.2) rolling In a beam  sea of < ' maximum wave slope  14°.  Figure 2  demonstrates : . that a reasonable stability range Is obtained, and ; the methods of Ref.[l] show the ship to be stable. - Now  decrease the damping coefficient by 25% (e.g. remove bilge keels) and the diagram alters In such   may be  found   - the a way that no Lyapunov bound  ship is now  unstable  (Plg.3).

4.1 Guidelines for the h  Function

(11) Guidelines for assessing the  implic ations for  stability  of any   given Lyapunov diagram.

If the h  function  Is to  have the required characteristics [l], i.e.

and h' > 0 g] for some region

> o

-8. < 8 < 8. h h

then It can be   shown that the  following cons traints must be  observed) a > 0  b > 0 c > a 

where c - f(8)

 8  will be defined   by and 

Formal experiments performed  on 8  ships confirmed  an earlier impression that coefficients a  and b,   although arbitrary, tend  to  iterate towards a narrow range of values when a  Lyapunov  bound  is  searched  for by   using the methods of section 2. The  results of these experiments may  be summarised

110.0 DEO

Figure 2   Ship No.5 Normal Damping

4

(i) a tends to lie in the  range (c - 0.1) < a < (c - 0.01) (11) b tends to lie in the  range 0.1 E-6 < b < 0.1 B-5

SIMPLIFYING THE CRITERIA

 Work is now in progress towards simplifying the technique in order that it may be possible to rigidly formulate a criterion for ship stability.  Any fixed criterion will be dependent on;

(111) a and b  tend   to  decrease with Increased excitation \ .

(lv) As a consequence of ( l i l ) 8. tends to n . Increase with excitation.

(i) Guidelines for the choice of the  coef ficients of the h  function, a and b.

- 66 -

4.2 Assessing the Lyapunov Diagram

If  Aj 'Are a, bounded by excitation lever, and 4>. curve

and  Aj • Area bounded by 0 axis and 4>, curve Then define a new. parameter A, where

.V

 A - r «  loox A

;

2

 As the excitation is Increased then A will naturally decrease until a critical value Is reached coinciding with the point at which the ship becomes unstable. Define  A

- Lowest value of A for which ship remains stable.

Figures 4 to 11 show the Lyapunov diagrams of 8 ships at this critical point.. Values of A were c •

obtained from these diagrams and.recorded In Table 1, along with the Stability Range. The general •particulars of the ships are given In Table 2. Ship

8

1 (deg)

No •

1 '': •'

2

.23.5 41.0 40.5 46.0 57.0 26.5 25.5 65.0

;•'

3 4 5 6 7 8

Table 1

Stability 92 (deg) 35.5 37.0 . 37*0 42.0 42.5 31.0 23.5 28.5

59.0 78.0 77.5 88.0 99.5 57.5 49.0 93.5

A

c (Z)

10.5 11.7

7.5 2.0 8.8 12.0 5.9

:

7.0

Parameters Obtained From the Lyapunov Diagrams of Figs.4-11

Ship ; No.

LBP

B

(m)

(m)

1 2 3 4 5 6 7 8

52.6 56.8 134.1 67.1 64.0 135.0 20.0 80.0

12.2 12.2 20.4 11.0 11.6 23.0

6.7 16.0

Table 2

T GM  A (m) (tonne) (m) 4.89 4.27 4.02 4.29 9.60 8.20 2.50 3.90

2122 1567 7912 2344 1532 16098

160 2706

1.72 0.70 3.78 1.26 0.78 1.70 0^75 0.38

f (s"1) 0.36 0.26 0.10 0.66 0.13 0,02 0.26 0.07

Ship. Particulars

It may be possible to develop a criterion for stability • in terms of stability rangé, but  par ame ter A is chosen here on the basis that •

c

-

'

i

simple geometry constrains it to be the more stable parameter.

0.80

0.60

 METR ES

70.0 GEO

Figure 7

Ship No.4

Figure 10 Ship No.7

0.80

0.80

 MET RES

 METRES

110.0 OEQ

Figure 8

Figuré U

Ship No. 5

Ship No.8

O.N  METRES 0.00

UNSTABLE

INDETERMINATE

STABLE

80.0 OEQ

Figure ?

Figure 12 Ranges of Stability Defined by Area •••-•' Criteria. •

Ship No.6

m 68 -

The task ia now to determine statistically a value for A applicable to any given ship. It can c  be seen fron Table 1 that A is not constant, but c we may define  A  A

- tower limit of A c2

- Opper limit of A c

The sample of 8 ships yields values of 2X and 12Z for A ând A respectively. Therefore, for c c l 2 any given 4> curve, a stability range in terms of the excitation on a vessel may now be defined (Fig.12). Examining Fig.12, the practical implication of this diagram for any excitation lever projected onto it is clear. Without recall to iterative techniques, any given vessel may be pronounced stable, unstable or indeterminate. Our aim must  be to work towards making the indeterminate rangé as small as possible. 5

CONCLUSIONS AND EXTENSIONS

 At this stage, iterative techniques performed either manually or (more preferably) by computer, have a significant role to play in the method.  More precise guidelines for thé choice of the h function and interpretation of the area criteria, will follow fron a comprehensive statistical study of. many ships. Such a survey may reveal a link  between, these two parameters and a ship charac teristic, such as the GZ curve. Progress is also being made towards extending this method to vessels in following seas. Here,  parametric excitation is regarded as the differ ence between the sagging or hogging GZ curves and the calm water GZ curve, Implying an excitation lever varying with 0 as shown in Fig.13.

' 

It'

Figure 13 Following See Lyapunoy Diagram

•-* 69 -

6

ACKNOWLEDGEMENTS

The author is grateful to the Directors of British Maritime Technology (BMT) for permission to publish this paper, and to the Department of Transport as the financial sponsor of the SAFBSHIP  project of which this work is a part. Thanks are also due to Dr. F. Caldelra-Saralva for his help and advice. 7 

REFERENCES

[l] CALD BIRA -SAR AIVA , F. 'A Stabi lity Criterion for Ships Using Lyapunov's  Method.' RINA International Conference on 'The Safeshlp Project: Ship Stability and Sa fe ty '. London, June, 1986.

Third International Conference on  Stability of Ships and   Ocean Vehicles, Gdarisk rSept.   1986   Paper 1.1Î

THE BOUNDEDNESS OF ROLLING MOTION OF A   SHIP BY  LYAPUNOV S  METHOD F.

Caldelra-Saraiva

 ABSTRACT

AN AUXILIARY FPHCTIOH

2

In the present paper, Lyapunov Theory ia used to obtain stability bounds for the  non-linear equation of pure-roll of a ship under the action of wind  and waves, Examples of application are given, that show that the results obtained   do not seen unduly conservative.

1

Let us assume' that g, fj, f 2 and f 3 satisfy the following conditions I (A)  g(9) continuously dlfferentlable  and odd  in I - [-a, a]j with  g(6) > 0 for S e (0,a) and   g(a) - 0. (B) f j O ) , f 2 (8) and f 3 (8) even and   contin uous in I.  Also fj(8) positive in that Interval and   fj(8) and f 3 (8) non-nega tive.

INTRODDCTIOH

 As a preliminary step in the study of the rolling motion of a ship, this paper will consider the determination of  bounds for the following second-order equation which describes under certain simplifying assumptions the pure rolling  and   motion under the excitation of wind   waves;

From these assumptions it follows immediately that g(0) - g(-a) - 0, that G(8) = ƒ g(s)ds exists and 0 is continuously dlfferentlable and also that f. (9) is bounded away from 0 and o in the closed interval I.

9 + f^e) e + f2(e) è|e| + f3(e) ê 3 + + g(6) - e(t) 

  dlfferentlable and  function  T|(8) defined  in I, we can construct a Lyapunov function V(9,6) as follows: For

(1)

This is a  generalization, to  include non-linear damping, of the simpler Llënard equation 8 + f(8) e + g(8) - e(t)

-

every

continuous

 V(9.9) - \ [6 + t)(8)]2 + G(8)

(2)

The substantial derivative will then be

The stability and   boundedness of the trajec tories that satisfy Eqn(2) have been studied extensively using Lyapunov meth ods. Early examples can be seen in Refs.[l,7], while exten sive surveys of more up-to-date results can be found   for instance in Refs.[8,9]. The assumptions ueually made are that 8.g(8)>0 for all 6*0 and 8 ƒ g( s) ds + « as 8 + •». In the case of the

V (8,9;e) - - f 3S* -  (T|f 3+f 2s)9 3 + + (T) , -f i -f 2 8Ti)9

+ [e + «(V-f^lê + n(e-g) where the dependency on 9 has been dropped, to

rolling motion of a  ship, however, these assump tions, are not valid. In, fact  g(9)  will change change sign at  values of 8 different from' 0;  moreover the interest is not so much in stability or boundedness per se but in boundedness within specified limits (lower than the first vanishing   zero)[lO]. We shall  point of g(8) different from  therefore have to devise a different approach.

lighten the notation, and s .5 sign(8).

In the continuation we shall also at times use V. and V for V(9(t),S(t)) and   v(8(t),ê(t);e(t)) when we are  particularly Interested in the dependence on time.  We can now prove the following. Lemma 1

For any 9* e (0,a) we- can select an

odd continuously dlfferentlable function n(8) and

The approach followed  In this paper will be similar to that of Ref.[ll] which studied    Eqn(2), though of course some of the simpler results that .could  be derived  for the linear-damping case are now lost. The basic result, the establishing of Lyapunov bounds, can however still be obtained. Examples of .application to the rolling motion of ships will be presented  at the end.

a positive 6* such'that:

(a )  r\\m < t x(fi) in I (b) |e |< 6*  implies *(±8*,6;e) < 0 (c) jej < 8* and   ^(8,8:0) > 0 imply 9 • Ô - 0 Proof

-71

+

-

Let us put .

v - xt + x 2 + x 3

'.3 X, - - t^ s 8

where

f,S\-T,f,S3

For future use we shall note at this-stage that if we  select 8' e (8*,a) for i| sufficiently small we can also have  P(r|) < 0 for all 8 e (-8',-8*) 0  ( 8 * . 8 ' ) and j n 2 ( 8 * ) + 0(8*) < G(8').  Then for some 6* > 0 sufficiently small, all 8 and all e of absolute value at most 6* we will also have V(±  8'tê;e) < 0.

ó f, ri

X 3 f (tl'- *! - «f2t))82 + [e + T)(t)' - tj)]» + +

  (e-g)t) + 0Î3TI If e'.- 0, then A • P(n) < 0 except at 8 - 0 . So if 8 *  0, V(8,9;0) < 0.  V(8,e;0) > 0 implies therefore 6 - 0. But then

and a - 3 3 /4 4 . It follows Immediately that, for any 8 and 8

X 3 - (t|' - t y )  8'

1* |3

 X l - - f 2 (8) |5| 3 s 0

and so we must have 9 - 9 - 0 . Aleo

ax,

.

-f--'-  58

For any 6* e  (0,a), 6 < 6* as defined in Lemma 1 and V defined using the corresponding tl(9), we can now define the correspondences

(40 + 3t,)

tJ' 

and ao for any given 8, X  reaches Its maximum for

9(6) - {(8,8) 1 J e I < 8* and, for some e, |e| < 6 and v(8,S;e) > 0}

1 < ^ f  (^3*„* -. 4 J3 3 4 3 3 „•) 4 - 0 X2(8,5) „• + i3 . 4 4 4

e'(6) - {(8,8) : 8* < |e| < a and, for some e, |e| < 6 and V(8,§se) > 0}

Let us fi nal ly consider X,.  Seen as a function   of 9, It  consists  of two second-order polynomials, one for 8 > 0 and one for 8 < 0. The discriminant la

and we can prove the following.

Lemma 2 A - e 2 - e  2[n(n' - tj '

2

- 

2 s n2f2] +

2

+ 4T)  (g -

o

T| f 3 )(t)'

- f x - s

A i s  itself  a  second-order polynomial two roots e 1 and e ,  satisfy et

e 2 -  '

'

P(r,)

'

'

•

T)2(V

•

For S e [0,ô*], 0 and e' are  upperseml-contlnuous  and 9(6) and 9'(6) are compact and non-void.

•

'

Proof The  proof Is similar for the two correspondences, so we shall just consider 9. Suppose 8 ± • 6 Q , 6  L < 6*, (8 1 t8 1 ) e 9(6 t ) and

f 2ri)

In e  whose

(Oj-.Ôj) • (9o»^o**   T h e n 6 0 * 6** F o r e a c h * there is e. such that $(8 .,ê. :e.) > 0 and e I < 6. < 8*, so there is a subsequence of e

- f x )  + 4n(g 2

3

tending to a limit e. Then e-l < 6*  and V ( 8 0 . 8 0 ; e 0 ) > 0 . Alao |8 0 | < 8* and, from Lemma 1 , 8 0 *.  ± 8*. so |e 0 | < 8* i.e. (8 0 ,Ô 0 ) e 9(6 0 ).

•

- a Ti f 3 ) ( j l ' - t Y  - sf 2  ti)

This est abl ish es upper-seml-contlnulty. Closedness of 9( 6) fo llows from the   same argument.

We shall now show'that we can  choose  T)(8)  odd and continuously differentiate such that t i ' ( é ) < f , ( e ) In I an d P ( n )  n e g a t i v e In [-8*,8*]\{0}. Since f   .Ms' positive in I, i t is clear that we can  choose  n(8) odd,  continuously differentlable and Incr easin g, such that T I ' ( 8 )  < f , ( 8 )

in I.

Then, if

n(8) is

The sets 9(6) and 9'(6) are non-void for any 6 e [0,6*], since (0,0) e 9(6) and (ia,0) e 9'(6). The sets are also bounded in 8, by definition, so all that remains is to show that 8 is bounded.

chosen

sufficiently small, the dominating term In P(t)) w i l l be  4 t ) g ( n ' - f  1 ) which I s  n e g a t i v e i n

It Is clear from the expression of V that if •4

[-e*,e*]\{o}.

•

#

f.(8). * 0 , the term -f- 8  dominates for 8 large In absolute value. It follows then that 9 < 0 for If f 0 but f, * 0  then the  term high J 8 -f , |S|   dominates and the same argument -f2s§3 applies. Finally, If f, • f • 0 the  highestorder term is (n' - f.) Ô which again Is negative

Then the two roots of A will be real and of opposite signs. We can therefore choose 6* < Mln{ e, , e } and if lei < 6*, then A < 0, ao X, < 0 and therefore V(± 6*,8;e) < 0.

72

fo r 9 la r g e tn ab so lu te va lu e. For any 9 e I and e e  [ - 6 * . 6 * ]  let us define § > 0 as e it he r 0 If V ( 8 , 5 ; e ) < 0 for a l l 5 or the modulus of the r oot of hig he st ab so lu te value If V(8 ,9:e) " 0 has a r e a l 9 r o o t . From what we sai d above It follows t h a t 6 U ex is ts always and Is fi ni te . Since roots are continuous functions of the coefftcents and 0 and e belong to compact Intervals, It follows that there U 8 suc h th at for a l l 8 e 1 and a l l e c [- 6*, 6* ], If |ê j > Ô ^ 0(8, Ô;e ) » 0. So, for any (8.Ô) e 9(6) U e' ( 6) , | ê | < 6 ^ .

Define t he neighbourhood of the or ig in Q ( ( 9 , 6 ) : | e | < 9* and V( 8. 9) < W(6>). Suppose th at ( 8 ( t 0 ) , 9 ( t 0 ) ) e e, | e ( t ) | < 6 for al l t > t Q  and, f o r s om e t * > t - we h ad ( 8 ( t * ) , ê ( t * ) ) 4 Q. Because Q Is clos ed and the tra je ct or y contin uous, t h e r e w i l l be t, in [ t Q , t * ) such th at ( 9 ( t 1 ) , § ( t . )) belongs to the boundary of Û, while ( 0 ( 0 , 6 ( 0 ) i  Q for a l l t E   ( t ^ t * ] . I f 8 ( 1 ^ e q ua l l e d 8 * we would have V > G(8*) > W(6), so I t must be 8( t ) < 8* and th e r ef or e V •> W(6). For some t e (t , t * ] arid a l l t E (t , t J we must th en ' hav e 8( t ) < 8* and V > V - W(6). It follows then from the definitions of W(6) and 6(6) t h a t V <  0 In ( t , , t _ J , and we reach a con tradiction.

Since 6(6) and 0'(6) are non-void and compact for ft e [0, 6* ] and V I s a contin uous fun cti on, we can define W(6) - Max(V(8i&) i (8 , 8) E  6(6)} W'(6) - Min|V(8,è) ! (8,8) E  Q'U))

Theorem 2

Given any 8* e  (0,a),  there will be

6 > 0 and 8* such that If (8(t0),ê(t0)) - (8*.8*) and

e(t) < 6 for all t > 0 then 8(t) < a-s for

and we can prove the following.

some E > 0 and all t > t„.

Leaoa 3

Proof

W(6) Is continuous at 6 - 0.

As stated in the proof of Lemma 1, there

is 8' E (8*,a) and V such that V(8*,0) <- G(8') and

Proof

Suppose 6, • 0 and (Ô..8.) E 6(6.) are

such that W(6 £ ) - V (8 1 ,ô l ).

V(±  e',è;e) > 0 for som e 6* > 0, al l 8, 8* < Jo| < 8' and lei < 6*.

Since |e i | < 8* and

follows that there is 6 E (0,6*] such that

IS.I•< 6,_, It follows that the sequence (8.,$.) Is • 1 '. MM 1,1 bounded and therefore has limit points. But from

W(6) <  V(9*,0).

Lemma 2 and Lemma 1 condition (c) It follows that

(8,8) E 9'(6) such that

all

such

limit

points

From Lemma 3 it

For such 6

there will be

are  ( 0 , 0 ) ,   thus W ( 6 ) - y [8 + n(8)] 2 + G(8) > G(8)

(8,,8.) > (0,0) and continuity of V implies W( 6 t ) ••  W(0).

> G(8') > V(8*,0) 3

THE B00NDBDHBSS OF TRAJBCTOaiBS There is .therefore 9* such that We shall now prove two theorems that answer

the following questions:

W'(6) > C(8') > V(8*,8*) > W(6)

Given a certain maximum desirable motion

Define

0*1, Is there a bound in the excita

o < f(8,e) : |e| < e' and v(e.e*) < v(e*,ê*)|

tions and a set of Initial conditions that guarantee that |o*

will never be

..exceeded? 2

Given any Initial position In the (open) Interval . I, excitation

is there a bound on the

that

guarantees

that

the

motion will never leave I?

Theorem 1

For any 8* e (0,a) there is 6 > 0

and à neighbourhood of the origin Q such that if tQ ,

S u p p o s e t h a t ( 8 ( t 0 ) , 8 ( t 0 ) ) - (8 *, 8* ) E Q, | e ( t ) | < 6 fo r a l l t > t - and f or some t* > t . , ( 8 ( t ) , § ( t ) ) i  Q. Then, as in the proof of Theorem I, It follows th at the re wi ll be t In [t f l ,t*) s u ch t h a t | e ( t 1 ) | < 8' and V t - V(8*, S*). There wi ll be th er ef or e a non-empty In te rv al (t ,t ] such tha t for t In such In t e r v a l V > V(8*,ô*) wh il e a t th e same time W' (6 ) > V > W(6) which Implies 0 < 0. We r e a c h t h e r e f o r e a contradiction.

then (8(t),8(t)) e 0 and  |e(t)|   < 8* for all t > tQ.

APPLICATION

Proof  Using Lemma 1, we can ob tain 6* and a function n(8) which will give rise to a Lyapunov fu nc ti on V( 8, 8) . From Lemma 3 I t fol low s t hat there will be 6 E (0,6*] such that W(6) <  G(8*).

The rolling motion of a trawler can be described by an equation of the form of eqn.(l) with

•- 73

fj(e) « .0267 s f2 (9) - .00655 degree

-1 -1 s

 \ / ^ 

f 3 ( 9 )  - 0.

and

g(9) - .279 e - .750 x io-2 e|e| + + .542 ji io"

3

e

A

3

.136 jt 10 4 e-.31| e |

\/y

A

WM liori fMX

•••»

tmt. co-mt

M.I I».»

IH PIMfl IKCUATIM •HIP IHCMID FW mut tntcacrpT ••.«*-11.M TMJ.  ftMM •17H.IT*. H

».»

+

HB

t. n

•iet M« KO

HO ITERATION REQUESTED

+ .132 x i o - 6 e 5 -

Fig.2

»-9.5 -2 - .466 x 10 e Jo I deg 8 5

where 0 is In degrees.

COHCLU8IOH8 AND BMBHSIOHS

The excitation satisfies

< 6 " 1.98 deg s~ 2 ,

which corresponds to

In'this paper we have addressed  the question

Then in Fig.l we see

of finding a Lyapunov envelope  for the motion of a

the corresponding regions  9(6) and 0'(6) (whose

system described  by a second-order equation, when

are   defined  by  crosses)  and the

the restoring coefficient can change its sign at

je(t)

wave slopes of 8 degrees.  boundaries

Lyapunov bound V(9,0)  - H(6) (defined  by the

 points

 pointed line) which Intersects  the  9-axle for loi

•

40  degrees.

different

from  the  origin.

He  have

 presented  a  method  for  constructing a  Lyapunov

Hé see  also the  result  of a

function and a bound  to the trajectories. He have

simulation starting from  the origin and   driven by

also found  on  application  of the method  to the

an excitation of  absolute value always 6, which

rolling, motion of a trawler that the bound looked

reached values   of 0 of  about 39  degrees.  I.e.

efficient.

within 2Z of the Lyapunov bound. Further F i g . 2  we

In

see

a Lya pun ov

studies   are

at

present

concluded which generalize Eqn(l)

bound

 parametric

 V(8,8) - W '( 6) (for a different  V   though) and a

to  encompass

  coupling with other and 

of motion.

 modes

simulation that, starting outside' the bound   (but

excitation

being

under the vanishing angle   of g(S) which is 94 6 

degrees) rapidly left  the Interval I.

ACKNOWLEDGEMENTS

I am    grateful  to the  Directors of  British

These applications of Theorems 1 and 2 show

(BMT) for  permission to  publish this paper,  and to the Department of Trade

 Maritime

us that the  bounds obtained  are not necessarily conservative.

Technology

as the financial sponsor  of the project of which this work Is a part. H

 \

M 1

"•. «•

ƒ 17 J

'*'*

K

1

>. Vt'

/*

V lU

II '

D •

i

K

 /

•

k"

1 vc^ V>"-*1L

X II

^«^ZT ^^ »*

•

1

^S"

'

•

•

•I«

 III

 JJJ  tsr^

G.E.H.

 /•

• • W

"^^a^S,

 Mr"

•

from discussions

BtO/S

.

•

i

This paper'has benefited

  correspondence with Prof. and 

Reuter to whom  I am    very grateful.

I

should also like to thank Mr. Stephen Phillips  of

:

%

BMT for his help with  the example.

R

"•

REFERENCES

* * • «• "-

\

CARTHRIGHT,   M.L. KO MVf tlOK t. M tMK i.n 1.0 i n n .  CO-OftM IH PMMf tXCITMIOfl HM (KIP  ( W I T »  r
Forced

Oscillations

In

 Non-linear Systems In LEFSCHETZ, S.  (Ed.).

•-•

Contributions   to the  Theory

of  Non-linear Oscillations. (Princeton University Press, 1950). Vol.l, pp.149-241.

.  *>••

NO ITERATION REQUESTED

Fig.l

A    Boundedness Theorem   for  Non-linear Differential Equations of the Second Order. Proc. Camb. Philos.   S o c , 1951, 47, pp.49-54. REUTER, G.E.H.

74

3

MIZOHATA,

S. and Y AMACUT I, M.

On the

8

Existence of Periodic Solutions of the  No n- li ne ar

Dif fe ren ti al

KNOWLES, I.

On Stability Conditions for

Second Order Linear Differential Equations.

Eq ua ti on

Journal of Differential Equations, 1979, 34,

x + a(x) x + 9(x ) - p(t), Mem. Coll. Scl.

 pp.179-203.

Univ. Kyoto, 1952, A27, pp.109-113. 9 4

ANTOSIEWICZ, H.A. On Non-linear Differential

Differential Equations. (Ed. R.  Academic Press, 1981, pp.421-429.

Equation of the Second Order with Integrable Forcing Term, J.

London Math.

STAUDE, U. Uniqueness of Periodic Solutions of the LISnard Equation In Recent Advances in

Soc, 1955,

Conti)

3 0 , pp.64-67. 10 5

LEVIN,

J.J.

and

NOHEL,

J.A.

ODABASI, A.Y.

Conceptual Understanding of

the Stability Theory of Schiffstechnik, 1978, 25, pp.1-18.

Global

 Asymptotic Stability for Non-linear Systems

Ships.

of Differential Equations and Applications to Reactor

Dynamics.

Arch. Rational Mech.

11

 Anal., I960, 5, pp.194-211.

ÇALDEIRA-SARAIVA, Solutions

F.

The Boundedness of

of LiSnard Equation

 Vanishing Restoring Term. 6

applied Math.

LaSALLB, J.P. and LBFSCHBTZ, S. Stability by Lyapunov's Direct Method with Applications. (Academic Press,  1961).

7

YOSHIZAWA, T. Second Method.

Stability Theory by Lyapunov's (The Mathematical Society of

Japan, 1966).

- 75

with a

IMA Journal of

(Forthcoming). .

Third International Conference on Stability of Ships and Ocean  Vehicles,  Gdarfs£,Sepf. 1986

wmw> Paper 1.12

 NUMERICAL CALCULATION OF FORCES AND MOMENTS DUE TO FLUID  MOTIONS IN TANKS AND DAMAGED COMPARTMENTS P. Petey

Abstract In view of simulating the ship motions in a seaway the liquid flow In tanks with a free surface is nur merically simulated. For the low fill depth case the shallow water equations are employed. The appearance of hydraulic jumps causes the ordinary difference-schemes to fail. Therefore Glimm's method is use d. For the deep fill case the free surface of the liquid remains essentially flat, since the greatest natural period of the tanks is much smaller than the period of the main excitation due to the rol ling

moti ons. A ve ry simple equat ion of

motion

is thus derived which can be easily integrated nu merically. The results obtained from numerical flow simulations are compared to experimental results and to analytical solutions. . 1.  Introduction

If we are intereste d in solving the' ship equa tions of motion in the time domain, we have to take account of the dynamic forces exerted on the ship .by the liquid in the tanks, in flooded compartments and also oh the ship deck, if it happens to become totally or partially awash. Since we are interested in computing ship motions for several ship geometries, seaway conditions and for relatively long periods of time, it is of para mount importance to use a model which, being accu rate enough, requires a relatively small computa tional effort. As a result of this consideration, this model represents a compromise of the conflicting requirements'for accuracy versus computer demand.

In the simulation, time is advanced in increments. The forces due to the liquid motions inside the ship at each time step can be added to other external forces (such as wave exciting forces, wind forces etc.), so that we obtain a complete time domain solution for both ship motion and internal fluid motion. The rate of flow of water through orifices in the ship hull »and through scupper s is estimated by straightforward equations, so that the volume of water inside the ship can be corrected at each time step  III. 2.Low fill depth case For the case in which the liquid fill depths are small compared,to the tank width, the velocity vector of a fluid particle is almost parallel to the tank bottom. We may thus neglect the velocity component perpen dicular to the tank bottom and assume that the par ticle velocity is independent of the vertical coordi nate. We investigate first the case in which the movement of the fluid particles is restricted to the yz-plane (f ig .l) . The liquid in the tank is allowed to slosh freely back and forth between the tank walls in response to a prescribed rolling motion. We com pute an one-dimensional flow described by the ve locity v( y) in the y-direction of a fluid particle relative to the moving frame attached to the tank and the liquid depth h(y) (measured in z-direction). We can write the so. called shallow water equations for this case (2) as a) Conservation of momentum in y-direction

The method is used as a subroutine in ship motion programs. It is suitable for simulation of large-am plitude motions of the ship which is considered to be a six-degree-of-f reedom system.

3 v+ v + f- 3h = f.. 3y 3t ay - 77 -••

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- 79 -

b) Conservation of mass

3t

3y

3y

where f_= g cos« - * J R + $y + 2$v f = -g sin* + {»y + $R

and g is the gravitational acceleration. At a first stage we solved equations (1) and (2) by using a traditional finite-difference integration scheme. The simulation did not remain stable, if the tank bottom becomes partially dr y, or if, in the case where the tank oscillates harmonically, the forcing rolling period and the natural tank period closely match (re sonanc e). Experiments show that in the resonance case a hydraulic jump is formed which travels periodically back and forth between the tank walls. The energy of the fluid particles crossing the Jump is not conserved. As a conse quence of the discontinuity in the liquid surface at thé travelling jump the derivatives of h( y) and v( y) with respect to y in (1) and ( 2) become infinite and the solution diverges. We use therefore Glimm's method /2 or 5/- to solve the above nonlinear hyperbolic system of equations. Both h( y) and v( y) are computed at points on a grid across the tank by "randomly" sampling ex  plicit wave solutions obtained from Rlemann (also known as dain-breaking) problems at each location on the grid at each time st ep. This sampling is performed so that momentum and mass are conserved in the mean. The mechanism by which Glimm's method propagates the solution to the equation is rather different from that of finite-difference schemes, as it requires many time step s for the cumulative effect of the sampling to give correct wave speeds. One feature of the results is that they contain a certain fluctuation due to the randomness of the method. Results are correct on the average only. No special handling is required of hydraulic jumps or for the case in which the tank bottom becomes partially dry. Interaction of the fluid with the tank top cannot be dealt with. The method requires a computational effort equiva lent to that of classical finite-difference schemes. If in addition to rolling the other 5 degrees of free dom are to be taken in account, we have to consider the flow both in x- and in y-direction. We introduce thus a third variable (velocity of a fluid particle in x-dlrection) for the description of the flow and use,

in addition to equations (1) and (2), the conserva tion of momentum law in x-directlon /5/. Chorin /4/ has shown how Glimm's method can be used to com pute complicated two-dimensional flow problems by using a so called splitting technique. The approxi mate solution to the 2-dimensional flow is construc ted at each time step from a combination of solutions to several one-dimensional flows in x- and in y-di rection. The method conserves both momentum and mass in the mean and exhibits a certain degree of randomness too. A detailed discussion is given in reference  161. Fig.2 shows a comparison between experiment / 3 / and simulation for the forced harmonic oscillation of an open rectangular container (resonance case). Fig.3 is the flow visualization. For very slow forced harmonic small-amplitude os cillations of an open rectangular container an ex  plicit solution 131 can be obtained from the lin earized approximation of equations (1) and (2). The numerical results (fig.4, sway force) seem correct, on the average. The deviations are due to the ran domness of the method. Fig.5 is the numerical simulation of a dam-hrenking problem. According to the analytical solution 111 a single shock, which travels with constant speed (approximately 664 mm/s) to the right, and a single rarefraction, which propagates with constant speed 700 mm/s to the lef t, are formed. The free surface of the liquid is depicted with a time increment of approximately 0.1 8. The mean speeds are correct. We notice that the shock remains perfectly sharp. Next we compute the following two-dimensional problem to check the validity of Chorin's scheme: a watertight bulkhead joins diagonally two corners of a square open tank dividing it in two equal regions filled with the same liquid but with differ ent fill depths (SO and 10 mm respectively) on each side. The tank is at rest. The bulkhead (the dam) is suddenly removed at time t=0, so that, in the central region of the tank, the previous onedimensional dam-breaking problem is repeated. Fig.6 shows the free surface of the liquid just after the removal of the bulkhead. Fig.7 is a ver tical section A-A across the tank, perpendicular to the removed bulkhead. The mean speeds are cor rect. A comparison between fig.7 and fig.S shows however that the two-dimensional shock is not per fectly sharp. Boundary conditions are treated by reflection tech nique. If the tank boundary is parallel to one of

Fig. 7.  Section A-A across the tank of figure g perpendicular to the removed dam. Water surface depicted in time increments of approximately 0.2 e.

ensH

*n° roll amplitude = S deg fill depth method

8'

Fig.8. Propagation of a small disturb ance in a cylindrical tank with low fill depth. The disturbance propagates out of the origin and is reflected at the tank wall.

o • 1 = tank length = 5 m b = tank width » 8 m hQ= still fill depth » b/5 T = natural period « 4.0 s R = distance to roll axis » 0.0

Fig. 10.  Comparison of results from the deep fill depth method with Glimm's method. The rectangular tank, Initially at rest, begins oscillating harmonically with forcing period 10 s. ;  v-V - 81 -

the mesh directions no problem arises. If it lies obliquely on the grid, it can be treated by a tech nique introduced by Chorin HI. We can however obtain reliable results , if we simply decompose the oblique boundary stepwise in the two mesh di rections. To check the validity of this approxima tion we picked a problem which is by no means tailored to the mesh geometry: an open circular cylindrical container (vertical orientation) filled with liquid. The initial circular free surface was at rest , when we introduced a small disturbance at its center. This small disturbance is propagated out of the origin in form of a ring with constant spe ed, is reflected at the cylindrical tank wall and returns to the origin. The numerical results are depicted in fig.8. The propagation speed is correct, on the average. The shape of the ring however is somewhat sharper in the mesh directions.

complicated tank shapes y' has to be computed prior to the simulation and stored. During the sim ulation this value is interpolated and the derivative of y' with respect to o computed numerically. A damping term could be easily introduced in (3). Fig. 10  shows several comparisons between this deep fill method and the low fill depth one for an open rectangular tank. The filling h. /b = 0.2 repre sents the upper limit of the range of validity of the shallow water method / 5 / . The tank, initially at rest begins oscillating harmonically with a forcing period of 10 s. 4.   References 111 H. Soeding, Damage Stability in a Seaway, (in German), Report No.429 Institut fUr Schiffbau, 1982 121 J. Dillingham, Motion Studies of a Vessel with Water on Deck, Marine Technology, Vol.18, No.l, 1981 131 Verhagen and Wijngaarden, Non-linear Oscil lations of Fluid in a Container, Journal of Fluid Mechanics, 1965  IAI  A. J. Chorin, Random Choice Solution of Hy perbolic Systems, Journal of Computational Physics 22, 1976 151 F.J. Petey, Calculation of Liquid Motions in Partially Filled Tanks and Flooded Compart ments (in German), Schiffstechnik, Vol. 32, No.  2, 1985 161 P. Colella, Glimm's method for Gas Dynamics, SIAM J. SCI. STAT. COMPUT., Vol.3, 1982

3.Deep fill depth case For tanks with deep fill depth the greatest natural period and the period of the main excitation due to the rolling motions do not closely match, since the former is much smaller than the latter ( fi g. 9) . As a result the liquid free surface may be oblique but remains essentially flat. Moreover, the ship motions are relatively slow, so that only the gravitational force is expected to play an important rote.

The influence of trim could be easily accounted for, but this would introduce undue complication in the present case. If the volume of liquid inside the tank remains constant during the simulation, the position of the liquid free surface can thus be described by only one parameter, the transversal 5. Author surface slope o. We determine the position of the center of gravity of the liquid mass in the Fernando Pet ey, Institut für Schiffbau der tank for several surface slop es. The curve so obUniversität Hamburg, Federal Republic of Germany tained repres ents the path along which this cen ter of gravi ty, at which all the liquid mass is as  sumed to be concentrated, is allowed to move as a hQ /b= 0.05 u changes during the simulation. The equation of «8 motion of the liquid free surface can be obtained Sei= 0.10 from Lagrange's equation   III: H • = 0.20 "SOB"

_£-

ay'/aa

si n( $ + a) co s( a) co s( 8)

2

= 0.30

(3) g _

a 0.50

8°

where y' is the position of the center of gravity o of the liquid in y-dlrection and 9 is the pitch angle. If the tank shape is rectangular, sp herical, ooo cylindrical etc., all terms in equation (3) can be computed explicitely in the program. For more - 82

hQ= still fill depth b = tank width 8.00

T 16.00

24.00

T

TANK WIDTH B CM!'

Fig.9.  Rectangular tank

 —I 32.00

40.00

Third International Conference on Stability of Ships and Ocean Vehicles,  Gdarisk.Sepl 1986 Paper 2.1

THE SAFETY AGAINST CAPSIZING IN RELATION TO SEAWAY PROPERTIES IN MODEL TESTS P. Blume

 ABSTRACT Significant wave height and peak period are not sufficient for the description of the severity of a seaway with respect to  capsizing tests. Therefore tests with 2  models were performed  In  different  to spectral shape, significant seaways with regard  wave height and peak period. In following waves the height of the center of gravity was varied  In order to find  the limit between safe and   unsafe.  As expected the correlation between significant wave height and stability parameters derived from the righting lever curve In smooth water for  the limiting KG Is not good. Heights of the Individual

wave group which hit the model are more suitable but In general It Is  difficult to determine them. However the correlation with statistical seaway data also can be Improved taking Into account the mean wave length and one grouping parameter beside the wave height. 1. INTRODUCTION There 1s a growing Interest In the development of new stability criteria which are  suitable also for modern ship designs In a  better way than the traditional requirements. It seems to us that up to now model tests are still the best approach to solve the problem with the smallest limitations In the  modelling of the real physical event.  But the -results of  such tests cannot be  seen as  absolute values (due tc a lot of   Influences and    arbitrary chosen  conditions);  they must be compared with results from corresponding tests with 'ihips of known  behaviour. Therefore 1t Is desirable to develop some standards for future capsizing tests. To guarantee the comparability of test results 1t Is necessary to control the conditions at these tests very carefully. Thereby the seaway properties  play an Important role. Significant wave height and  peak period alone  are not sufficient  for the description of the seaway with respect to capsizing.-' The investigation reported here was planned with the

aim to get some Insight Into the problem. It was sponsored  by the German Ministry of Research and Technology. Possibly the results can be taken as a first step to a standardization of  capsizing model tests In Irregular seas. In an earlier Investigation four different ship   A,B,C and D) were tested. First  models (named  results were presented   at the Stability 82 Conference [l]. As a result of further work a  proposal for stability criteria valid  for a distinct group of ships was derived which takes care of Individual geometrical parameters of the ships [2]. The first experiences with this proposal will be reported

during

this

conference

by  Hormann  und

 Wagner. For the present Investigation the  models A and C were used. Model A was tested  In 12 additional seaways whereas the program for model C was kept shorter with 5 seaways. Both models were tested at one draught on desired relative courses of 0 and 30 degrees. The test procedure was the same as in the earlier Investigation, all  tests were repeated several times up to at least 10 tests at the same conditions.

2. SELECTION AND ANALYSIS OF THE MODEL SEAWAYS

- The model seaways were selected with regard to the spectral shape, the significant wave height and the peak period. There are one group of  sharply  peaked spectra with JONSWAP-shape, peak enhancement y - 5> of factor another group Pearson-Moskow1tz-type  and one spectrum  as  wide as  possible. These spectra are named  J, P and B respectively with two numbers which Indicate Increasing wave height and  peak period with Increasing number. The control signals for these seaways were computed using 560 harmonic components. Thereby for the seaways with different heights but the same peak  period   in  both groups only the amplitudes of  the control signal were changed. A    comparslon of  the analysed data for the J-seaways with  the

- 83 -

corresponding data of the similar seaways I, II and III of the ealler Investigation shows differences which possibly can be attributed to the different number of components. The control signal for  the seaway I, II  and III was composed of 20 components only. So finally one additional seaway named  J 2.2 100 with 100 components was Investigated. F1g. 1 and 2 show the spectra of the model seaways. The curves are smoothed mean curves of 3 to 6 measurements.  All seaways were measured  at two positions using sonic,resistance and capacitive probes.  The distances from the wave maker were 54 m and 86 m. The time records were statistically analysed. Then spectra were calculated using a frequency resolution of 7.5% of the peak frequency and some spectra  parameters were determined. Beside these standard procedures the main analysis was done with regard  to the grouping  properties. The envelopes were analysed   and the values J  to  Goda [ 3 ] were 1 and J ?   according P20-10" 3

determined which give the mean number of successive waves 1n a group which are higher than a certain value and the mean number of waves from one group to the next. Further the SIWEH-function (Smoothed Instantaneous Wave Energy History) by  Funke and  Mansard [ 4 ] and  a grouping factor GF  were calculated. Also the correlation coefficients r HT for wave height and period, $ H m  for two successive wave heights and  *TT. for two successive periods Introduced by Rye [ 5 ] were determined. Finally data from the wave group which the models actually did  meet 1n each test were determined from video recordings, mainly the wave heights H and the crest heights n' above the mean water level. These values then were averaged over three successive waves.  All results cannot be given here. Therefore all data for the same seaway from the measurements at the two positions and with different wave probes were averaged. The main data are compiled In Table 1 which also contains the results for the seaways I, II and III of the earlier Investigation   [1,2]. The mean wave height H or the mean height j\' of the crests of the wave group relative to the .! significant wave height 4N 1£ depends on the type of ; the spectrum and on the wave height Itself as it 1s ; shown 1n Figure 3. The wave groups become relative 1 higher with Increasing peakedness of the spectrum  ' and decreasing significant wave height. Rye 1s in favour of the not undisputed opinion that the grouping properties only depend on  the spectral shape [ 3] with Increasing tendency at Increasing.peakedness of the spectrum. He explains different conclusions from other authors as well as the large scatter of the analysis results which can lead  to this other view with shortcomings  of  measurements or simulations and analysis methods as

J 1.2

J 2.2

3 4 2 Fig. 1 Spectra of the J-.seaways

P O

^ T l ^ ^ t . r 1 — -J^-2

 _\ <£*_•_-„M.«-

D74-*

B2.2 0 \

A) 2.2-100

4Vm7[nn] 0.5-"

2

3 , * F1g. 2 Spectra of the P- and B-seaways

300

400

500

600

Fig. 3 Relative mean wave heightfland crest height n' of the wave groups as function of the significant wave height

5

- 84 -

Table 1  Averaged data of the test seaways seaway

J  1.2

J  2.1

J  2.2

J  2.3

J  3.2

J  2.2 -100

P  0.2

P  1.2

P  2.2

P  2.3

P  3.2

B  2.2

HI

4/n£lm]

0.330

0.384

0.376

0.387

0.472

0.401

0.342

0.374

0.410

0.419

0.477

0.383

0.413

0.549

0.613

Tp

2.50

2.09

2.49

2.92

2.52

2.48

2.57

2.58

2.58

2.97

2.60

< 2.62 >

2.16

2.35

2.49

0.328

0.399

0.370

0.380

0.471

0.393

0.338

0.375

0.412

0.412

0.482

0.366

0.418

0.499

0.601

0.551

0.622

0.631

0.630

0.675

0.555

0.669

0.728

0.638

0.701

0.840

0.554

0.599

0.681

0.814

R 

(ml 0.720

0.787

0.850

0.797

0.873

0.603

0.703

0.667

0.717

0.687

0.773

0.625

0.860

n'  

Tm] 0.433

0.487

0.507

0.500

0.583

0.350

0.443

0.483

0.463

0.420

0.540

0.413

-

h 

1-) 0.0433 0.0703 0.0489 0.0411 0.0603

0.0550

0.0537 0.0559 0.0601 0.0519 0.0676

0.0601

0.0726 0.0744 0.0824

1.029

1.113

1.125

1.306

1.168

1.027

0.954

1.194

1.250

W


VL PP*I-.Î 1.347

0.938

1.315

1.576

1.310 - 1.045

"•

• -

-

II

I

0.579

Qp

M 3.80

4.24

3.82

3.42

4.05

4.14

2.32

2.43

2.52

2.17

2.69

1.98

4.84

4.81

4.58

GF 

(-) 0.881

0.823

0.906

0.879

0.884

0.716

0.856

0.940

0.904

0.739

0.850

0.663

0.721

0.676

0.716

* H H t l - » 0.529 J,   (-1 1.86

0.497

0.532

0.531

0.529

0.531

0.278

0.281

0.332

0.238

0.349

0.131

0.233

0.222

0.209

1.81

1.61

1.53

1.85

1.37

t.38

1.33

1.43

1.34

1.46

1.20

1.30

1.17

1.44

there are  limited length of records, Insufficient digitalIzatlon frequency and   too small number of components. The  shape  of the  spectrum   can be described  by the peakedness parameter Q  Introduced "by Goda. This value as well as some grouping  parameters now were averaged over all measurements within the three groups of spectral shape J, P and 1.2-r 7

B. Thereby only the seaways composed  of 560 components were used. The mean values and    their standard deviations are given In  Table 2 whereas In Figure 4 the mean values are drawn as  function of Q_   In comparlslon to the results of Rye derived from numerical simulations. With the exception of r U T the HI

results of the tests and simulations are more or less similar and of the same tendency; thereby one has to  keep 1n mind the wide scatter expressed by the standard deviations. The best agreement can be recognized  for the correlation factor *HH, for two successive wave heights. Tab.2-  Averaged grouping parameters and   their standard deviations

Spectr. J0NSWAP,y - 5 mean S GF 0.878 0.0539 5.78 0.579

VV r

HT

*HH1 *TT1

V

Pears.-H.-Sp mean S

0.847 4.80  0.333 0.0969 0.314  0.525 0.0576 0.284 0.192 0.0883 0.236 3.86 0.262 2.37 

0.0860 0.504 0.0984 0.0575 0.0756 0.209

wide spec mean S 0.663 3.90 0.378 0.131 0.126 1.98 -

In Figure 5 the dependency of the grouping  parameters from  the number n of the  harmonic components Within the control signal for the wave  maker Is shown. Here the values have been averaged over 6  measurements of the seaways I, II and III with only 20 components, 4 measurements of seaway J   2.2-100 with 100 components and 21 measurements of all J - seaways with 560 components. In spite of an decreasing Q - value the grouping parameters and the ratio of  maximum  to significant wave height Increases with Increasing number of components. Fig. 4  Grouping parameters as function of the  peakedness parameter - 85 -

1.8-, . w max (O) ç w sig GF (D) 1.6 *HH1 (V) <=w max max *TT1 1.4 1.2-

-6

1.0-

-5

0.8-

-4

0.60.4-

-1

0.2-

0 50 100 200 500 1000 10 20 Fig. 5 Grouping parameters and maximum wave height as function of number of components

3, MQPE1.S M P  TEST PROCEDURE  As already already menti mentioned oned two models models named  named A  and  and C from an earlier Investigation have been used again. The main dimensions dimension s of the models are compiled in Table 3 and the Figure 6 shows their cross sections. Tab.3

LPP [ m ] LOA [ m ] B 

[m] T  [m] D  (m) 1 V t m] CB CWP

LPP/B B/D B/T D/T

Main dimensions of the models model A

model C

4.821 5.180 0.821 0.293 0.382 0.783 0.675 0.816

4.938 5.272 0.875 0.344 0.503 0.985 0.663 0.849 5.64 1.74

5.87 2.15 2.80

i.3b

1.74 1.46

The models were tested at one draught only on desired courses of 0 and 30 degree relative to the waves (0 degree means following waves). During all all tests a static heeling heeling moment was acting on the  model  model producing producing a heeling heeling angle of 2 degrees. The free running model was hand operated by a helmsman  being  being accommo accommodat dated ed on a subcarrlage 1n front front of the  main carriage. carriage. The model model was connected  to this subcarrlage by flexible cables for energy supply and

ship C Fig. 6  Cross sections data transmitting. Carriage and subcarrlage followed the model so that no considerable forces acted onto the model. mod el. Due to the restricted width of the tank the e  model had to be  steered  on (18 m) th zig-zag-course's at the tests with 30 degree relative course. For each seaway a  starting delay relative to the start of the wave maker was determined at the first trials and then kept constant so that the  model  model always met the the same same wave wave group. group. Duri During ng the tests these wave groups were recorded by a  video camera which was directed at the opposite tank wall. The same wave shape was recorded at all all tests Ina the e position of the model  particu  particular lar seaway seaway but but th model relative to the crests differed from test test to test a little bit due to unavoidable changes 1n speed and course. So the main wave crests were  met in different phases of their development. This fact contributes essentially to the  scatter  of the  measured  measured data data and and 1s the the reason reason tha thatt a large number of runs 1s needed for the same condition condition with respect to seaway, desired course and    load condition. the e wave From these video recordings also th heights H  and heights n' of the crests above the  mean  mean water water leve levell mentio mentioned ned 1n the the sect sectio ionn before before were determined. Of course one cannot expect the same accuracy as for direct measurements. Thereby - 8 6 -

 fi' of the crests turned the height fi turned out to be more reliable and more Important. the different The tests were performed  1n the  different at  both courses for two to  to  four seaways and at  different heights of the center of gravity In order to find the limit between safe and unsafe. For a  better statistical reliability a reliability  att least 10, partly  at the same conditions. up to 23 runs were conducted  at condition s. So altogether nearly 1000 single runs were necessary during this Investigation. the   judgement of thé thé   safety against For the capsizing the remaining area E R   under the smooth the measured water righting lever curve beyond  the  measured  the  maximum heeling angle $ m x  was used. From  the Ep-values of each run the mean value E„ and  the  the standard standard deviation s deviation  s were calculated from all all tests at the same condition. For a  safe ship 1t 1s  by  three required that the mean value diminished  by times the standard deviation Is equal o equal  orr larger than zero:

The stability parameters parameter s of both ships can be  made comparable by multiplying .the them with the hull hull form factor C which has been Introduced In  [2].  The factor Is defined  by the equation T-P' T- P' "l/T "l/T -Êfi_ i/Ta i/T a cL _  " B'  f KG CWP V  X Further details can be found In [ 2] or In the contribution by Hormann and Wagner. For the base length L Q here a model value L Q - 100 m /28 - 3.57 m was used (28 Is the original scale of mode modell A ) . Depending on the height height of the center of gravity the hull form factors differ at  at  the same same model as the case may be, but thé factors for model C model  C are much higher than for model A:

4. CONCLUSIONS FROH THE TEST RESULTS

C = 0.1274 y - i -

 model  model C:

C = 0.1761

V£

with KG/T > KG/T > 1  1

 After the multiplication multiplication with the hull hull form form of both models don't factor the stability parameters of both show a significant difference for the same seaway. Is supported by the   t-test on differences (oh This Is supported The  t-value can be  be  calculated easily from  the  the differences d, their standa standard rd deviations s deviations  s   and the number n of tests (here 8 pairs):

(ER  - 3 s) > 0 .

 ER  - 3 s Figure 7 shows an example where £ R and  and  E are drawn as function of  of  the metacentric height. From these graphs the limiting limiting values of GM Q can easily be found. In the Figures 8 and 9 and 9 the righting lever curves at the limit between safe and unsafe for the different differe nt test seaways are shown for both  models.  mode ls. From these curves stability stability parameters can be derived which are are compiled  compiled  1n  1n the Tables 4 and 5. and 5. Here also the results from the earlier Investigation with seaway I, II and III are Included.

 model  model A: A:

t =

 V ^

Here the

values t - 0.500

and t

0.886 were

the differences of  of  C*GZ„ and determined   for the differences C*E C* E Q respectively which both are to be compared with and  3.499   3.499 for a level of the values t Q  »2.365 and  significance of 5% and and 1% respectively. respectively. The smaller t  clearly Indicate that there Is no values of t  the   weighted the significant difference between stability parameters parameters of both both models contrary contrary to the unweighted parameters paramete rs (for which t - 8.073 and t and t 5.912 were determined). That means the hull hull form factor works work s very well and the relative order  of  both models with respect respect to the safety against capsizing I capsizing  Iss always the same Indepent from from the used seaway.  Now correlations between the weighted weighted stability stability of  seaway parameters were  parameters and a set set of  calculated  by linear regression. The The   finally be  taken selected set of seaway parameters parameter s x can be  from Table 6  which contains the results  results  of the ! regression. There are given the mean value y and the standard deviation S of the stability stability parameters, of the regression Un the coefficients a and b and b of the regression  Un e y - ax + b, the standard standard deviation S x y  relative to the regression regression Un U n e and the correlation correlation coefficient fS e - S 2 '

 Itt was learned From the earlier Investigat Investigation ion I that from the considered stability parameters  the  maximum rlghtlmg lever GZ_ and the total area seems to be to be most E 0 under the righting lever curve seems appropriate for the discrimination between safe safe and unsafe. Therefore the following considerations are  the restricted restric ted to these two paramete para meters. rs. Cer.t Cer.ta1n a1nly ly the wave height Is the most Important Important seaway seaway parameter. In the Figures 10 and 11 the lever G ^ and the area E Q are shown in dependency depende ncy of the significant wave height 41^5^. 41^ 5^. The large scatter clearly Indicates that the significant wave height alone  alone  1s not sufficient to describe the severity of a seaway with w ith regard to the safety against capsizing. For example the necessary lever Gi^ In a JONSWAP-seaway JONSWAP-s eaway 1s about 1.5 1.5 times larger than 1 than  1nn seaway B 2.2 a 2.2 att the same Is  even wave height. For the area E 0 this . ratio Is  1s  true for both models, but the about 2. This 1s  but  the values for the model C model C are much lower than for than  for  A   model  model A    due to the more favourable form of this  model.  model . - 87

:-fy

SiV

 As expected the best correlation correlation coefficients at the single seaway parameters In the upper part of the table can be found for the crest height n'. By n'.  By adding the mean wave length ratio and mainly  the -

Table 4 Stability parameters at the. the. limit between safe and unsafe for model model A

KG

seaway

GZ

GZ

E

E

E

E E m o 30" E40 30 Ho [mm radj [mm rad] [mm rad] [mm rad] [mm rad]

[mm] [mm]

m [Grad]

[Grad]

18.7

19.9

34

56.8

5.83

9.26

3.43

7.31

12.35

24.3

26.4

26.8

37

63.7

7.40

12.01

4.26

10.49

18.57

52

22.8

23.3

23.9

37

61.0

6.77

10.90

4.1 3

9.45

15.87

283

68

30.8

33.5

33.7

38

71.0

8.90

14.66

5.76

13.22

25.37

301

50

21.8 21. 8

22.0

22.7

36

60.0

6.49

10.41

3.92

8.84

14.88

296

55

25.3 25. 3

25.1

25.4

37

62.7

7.174 7.174

11.58

4.41

7.84

17.51

292

59

26.3

27.7

28.1

37

65.0

7.71

12.53

4.83

11.23

19.68

P 3.2

286

65

29.3

31.6

31.9

38

69.2

8.41

13.88

5.46

12.51

23.29

B 2.2

308

43

18.3

17.5

18.8

34

55.5

5.49

8.70

3.22

6.77

11.39

I, II

291 291

61 61

26.8

28.6

28.9

38

66.5

7.86

12.86

5.00

11.96

20.71

III

300

52

22.5

23

23.9

37

61.0

6.79

10.86 '

4.07

9.57

15.75

[ran]

"»o [mm]

30 [mm]

40 ( mm ]

J 1.2

306

45

19.4

J 2.2

294

57

J 2.3

299

J 3.2 P 0.2 P 1.2 P 2.3 J 2.2 -100 P 2.2 J 2.1

,

the limit between safe and unsafe for model C Table 5 Stability, Stability, parameters param eters at the seaway

KG [mm]

GM [mm]

GZ

GZ

30 (  mm]

40 (  mm]

GZ m

m [Grad]

»o ( Grad]

m [mm ]

E

E E E E 30 40 30"E40 m o [mm  rad] (mm  rad] [mm  rad] [mm  rad} [mm rad ]

J 2.2

363

25

17.6

21.8

23.8

48

71.8

4.25

7.61

3.36

10.84

17.11

J 2.3

371

17

12.7

16.7

17.8

47

66.8

3.21

5.79

2.58

7.93

11.81

J 3.2

361

27

17.7

23.1

25.2

49

73.6

4.51

8.09

3.58

11.76

18.38

P 2.2

368

20

14.2

18.6

20.0

47

70.0

.3.60.

6.47

2.87

8.85

13.81

B 2.2

376

12

10.2

13.4

14.4

46

63.4

2.56

4.62

2.06

6.10

8.98

I

369

19

13.4

17.8

19.7

47

68.0

3.47

6.16

2.72

8.50

12.84

II

371

17

12.8

16.9

18.4

47

67.0

3.25

5.81

2.56

8.00

• 11.94

in

374

14

11.3

14.7

16.3

46

64.5

2.81

5.09

2.28

6.72

10.06

desired course E R;

C„-3 J (en

I

red)

P1.2

:

•

f

• 

*

0" o

30° O

X

A

h ! R -3S

' R! E R -3.

10

20

30

40

SO

Fig. 8 . Model Model A

F1g. 7  Determination ofGM  Q at the limit between safe and   unsafe

10

20

30

SO

Fig. 9 Model Model C

 in calm water at Righting lever curves in the limit between safe and   unsafe - 88 -

40

Table 6  Results of the regression regression analysis  — — — — — —

stability  parameter C 6Z

for   4VÜ- , ^ . 1 g , ^  N

23

y

3. 309 30 9 mm

'

0.559 mn

seamy  parameter (x)

8

[-1 i   l   H  i  

b f mm]

ni

TOX

fori?, n' 19 3.345 mn 0.591 0.591 mm

for   fl, n'

^ « V ^ i g . ^ M x 23 2.269 ran rad 0.508 0.5 08 ramrad

r

, W 1-1

0.329

0.508 0.491 0.413 0.428 0.423

0.416 0.478 0.674 0.690 0.697

0.007974

0.650

0.360

0.766

0.009174

1.115

0.314

n' • f(VL)

0.006804

0.153

n*-   »Hill"* ' f ( V L )

0.007682

max • « * " - > *w  ma

a I  radl    i

b [mm radl

19 2.307ramrad 0.529 0.529 nm rad S

r "l

mm rad J

[-1

1.206 0.961

0.454

-0.286 -0.620 -0.313

0.381 0.395 0.390

0.387 0.447 0.660 0.665 0.667

0.007007

-0.068

0.341

0.741

0.827

0.008178

0.313

0.296

0.812

0.405

0.727

0.005929

-0.475

0.374

0.708

0.562

0.319

0.842

0.006782

-0.149

0.295

0.830

0.004039

0.706

0.430

0.639

0.003587

•0.044

0.396

0.625

S» max • VHM1

0.005559

0.511

0.311

0.832

0.005015

-0.256

0.286

0.826

^n«x.*HH1,/4'f
0.006310

0.2Jâ

0.270

0.875

0.005686

-0.501

0.251

0.869

S«  sig *•*   max

n 4fi£'Gf   -f(VL) . *^^T" • H M * , / * -

*c*/i:>

2.050 1.770 0.438

-0.048

   I    I    I                 I

0.468

fU/L) = exp(-0 exp(-0.5 .5 (X/L-1 (X/ L-1)) 2 ) accuracy of the single measurements also  1s not grouping parameter « . the correlation could  be  be higher an  cannot  cannot be Increased  In a  simple way due  and  d  Improved  as  as shown In the lower part of the table. to the  Influences which are not not under und er  the plurality of Influences The mean crest height n' of the  actual control. encountered waves  ca  cann be determined only with a Summarizing one can state the following from larger effort. Therefore the crest height   Is not a this Investigation: good choice for the comparison of routine tests.  length  length - with regard  to the seaway the height an  and  d    Instead  of  of n' the wave height c  „„„measured  at  at one [  Instead  or shape of the actual actual encountered wave crests  are (ore (ore more) more ) stationary stationary positon(s) positon(s) can be used with of Importance,  bu  butt these data can be be  measured only about the same  same results. with larger effort. The ratio c w m a x /t w s1 and *HH) are dependent - two succeeding high waves raise  the  the  danger of from spectral shape  and   number  number of components with a  and  capsizing considerably,  bu  butt from  a higher number of similar tendency. So the maximum wave height in the waves doesn't follow a  significant additional seaway parameter also can be replaced  by  by  4V~nT and Increase. the ratio can be  Included  In the dependency from r r ne r a c t 1 c a use *HH1 ' ' ° * P  therefore the following l properties become -  herewith the grouping  paramete  parameters rs offere offeredd themselve themselvess for the description of Important. An irregular seaway  fo  forr  capsizing tests the severity  of an Irregular seaway with regard  to should contain sufficient high and  and   conti nous  wave the safety against capsizing: groups. Therefore  th  thee spectrum should  be not to wide. For the classification of different seaways seaways P with regard  to  to their severity two parameters are  mo =4 l f %  *HH1 1/ 2  exp(-0.S(X/L - I)  I )2)  propos  proposed ed which which are composed composed from wave height, mean ^ x = ^ m a x * H H 1 , / 4  exp(-0.5
- 

8Z n [im]

20 -

.30-

15 •

; -+

,-+ teawy-typ J ?  B  I-III mode modell A A O O + mode modell c A • • +

10-

10 . seawy-t seawy-typ yp J P B M I ! node) node) A A o • + mode modell C A • • +

«/ïCIwl

tfn?   [mm] 500

300

400

500

Fig. 11 Area Area E at a t the l i m i t betw betwee eenn safe and and unsafe .

Fig. 10 Righting lever GZ_   10 Righting  at  at the limit between

4-

'•«ml"""!

/ 

O  y O 

seawy-typ seawy-typ mode modell A mode modell C

sumy-typ J P B Ifl I I mode modell A A O O + mode modell C A • • +

,

-fe^ tot tot L / S

AX

J P B HUI A o O + A • • +

^„„• ,/4 e«p(-0.5()i/L-1)M

«Pt-O.SO S/l-D») (mm)

150

/l0*

y

y

^* HH 1

/

A

+ /

4yl

600

300

300

Kig. 12  12 Weighted lever GZ m   m  as function of the seaway parameter P

[m] 700

 13 Weighted lever GZ_ as function of Fig. 13 the seaway parameter P C E0[mmrad)

/V

A y\l ï o i /   °

3 -

yto«

 /

•

2 -

+ y +/ y rr.

<&>>''

y

/

'

/#X

yy+y y

 //  v

«eaua «eauay-t y-typ yp J P B H U I ondel A A o O + model model C

A O •

1 -

+

  expt-O.sU/L-l)1) (im|

in 

*f%* 

150

200

250

&

> seawy-typ seawy-typ J P. B K i l l mode modell A A O O + mode modell C A • • +

™" o -

300

400

Si «ax «ax '

«xpi-O .SiX/ l- I)«) lirni) lirni)

500

F1g. 15  15  Weighted area E as function of the seaway parameter  ?„,„

Fig. 14  14  Weighted area E as function of thé seaway parameter P ^

- 90 -

 possible due t o th e  stochastic nature Influence factors.

of many

 ACKNOWLEDGEMENT The support of this large Investigation by the  Mi nistry of Research and  Technolology of the Federal Republic of Germany Is gratefully acknowlegded.

 NOMENCLATURE B beam C hull form fact or CB block coefficient CWP waterplan e coefficient E 3 0 ,E 4 0 ,E _  area under  t he righting l ever curve up to  30 , 40 deg. and the maximum Eo total area unde r  the lever curve area beyond « m a x grouping factor metacentric height righting lever maxim um righting lever h = \ g ""i-iM ïrffir significant steepness H wav e height In a wave group J. mean number of waves In a group KG height of center of gravity above base L,LPP length between perpendiculars LOA length over  all variance of the seaway RI  P__ » Pmax seaway -* rparame ter  mo E R  GF GM Q GZ GZ

Q r •HT

S, s S (to) T T T  V y r  

i  v

w max

\ 

peakedness parameter correlation coefficient correlation coefficient for wave height and period standard deviation

spectr al density draught peak period mean period  of  SIWEH volume displacem ent enhancement factor mean of the upper 1/3 of wave heights maximum wave height

mean wave length $ max maximum heeling angle v TMi' * n im  •'corral, coefficient for 2 successive *uu4» wave heights and periods respectively circular frequency

BEFIBEMS 1. P.   Blume,   H.G.  Hattendorff, An Investigation on  Intact Stability of Fast Cargo Liners, II. International Conference  on Stability  of Ships and Ocean Veh icles. Tokyo, Octob er  1982 2.   Report on Stability and   Safety against Capsizing of Moder n Ship Pesi ons . International  Ma ritime Organization. Paper SLF/34, su bmi tte d  by the Federal Republic of Germany 3. Y. Goda,  On Wave Groups, Conference on Behaviour  o f Offshore Structures  /BOSS).  Trondhelm, 1976 4.   E.R. Funke, E.P.D. Mansard, On the Synthesis of  Realistic Se a  States In a Laboratory Flume,  National Research Council. Hydraulics Laboratory, Canada, Report LTR-HY-66,  1979 5. h\ Rye,   Ocean Wave Groups, Th e Norwegian Institute  o f Technology. Univ. Trondhelm, Report  No. UR-82-18, 1981 6.  E.L.  Crow,   F.A.  Davis,   M.W.  Maxfleld, Statistics Manua l. Dove r Publications. Ne w  York, 1960

Third International Conference on Stability of Ships and Ocean Vehicles,  Gdarisk, Sept   1986 Paper 2.2

EXPERIMENTAL STUDY ON PURE LOSS OF STABILITY IN REGULAR AND IRREGULAR FOLLOWING SEAS  N.

Umeda, Y. Yamakoshi

 ABST  ABSTRA RACT CT

wave to predict safety safety in irregular following following seas. However, he did did not make a stoc stochast hastic ic pre

The authors undertook tö study the pure loss of stability of a ship travelling in following following seas. We measured measured the stabilit stability y of a model ship

sentation of stability stability itself, itself, the capsizing capsizing de  pend  pends s on the the succ succes essi sive ve time time of stab stabil ilit ity y loss loss. .

towed towed with a heel angle in regular regular waves. Ihe

Thus Grim also also propo proposed sed tó use the the time time mean of

results were then compared compared with hydrostatic com  puta  putati tion ons s whic which h to some some exte extent nt were were able able tó prepre-

the effective effective wave. However, However, this notion notion is not directly connected with the actual occurrence occurrence of

diet such measure measured d values. Further, Further, He confi confirmed rmed

capsizing. Krappinger[6] computed computed the probability probability of capsizin capsizing g by making use of Grim's effective effective

that that the relation relation between between wave and stabi stability lity' ' is non-linear. thus one cannot cannot easily easily predict predict

wave concept. Further, he compared this results

stability stability in irregular irregular waves.

with the results of model experiments. experiments. Nonethe-

To

bypass bypass this difficulty, difficulty,

we made

a

' less less he was unable to clarify the validity validity of the

stochastic representation based on the concept of

effective wave concept concept because of the difficulty

Grim's effective effective wave: an irregular irregular wave profile profile

of carrying carrying out capsizing capsizing experiments. experiments. On the

is approximat approximated ed by a regular regular wave which is called called

other hand, Kastnert?] carried out time domain

"the effective effective wave." Moreover, we carried carried out . our model experiments experiments in irregula irregular r waves. Ihe

simulation simulation based on hydrostatic theory. Further he discussed the spe spectra ctra and probability probability distri

 meas  measur ured ed valu values es of stab stabil ilit ity y were were simi simila lar r to those predicted by using the hydrostatic computa tion and the concept concept of the effective effective wave. How ever, there are some difficulties difficulties in utilizing, utilizing, this method of prediction prediction for practical purposes. 1 ÏNIBODUCTICN The pure loss of stability stability in followin following g seas

 buti  bution on of stab stabil ilit ity y vari variat atio ion. n. His His resu result lts s were were not unfruitful; however, the procedure cannot easily be employed employed for practical purposes. Re Re cently Helas [8] proposed a stability stability criterion  base  based d on the the effe effect ctiv ive e wave wave conc concep ept. t.  We the the auth author ors s thin think k that that Grim Grim's 's effe effect ctiv ive e wave concept concept is an excellen excellent t one because stabili stabili ty variation variation are non-line non-linear ar phenomena. phenomena. Thus we attempted attempted to apply this concept by making model

is an importa important nt factor factor of the capsizing capsizing of ships. The transverse transverse stability stability of a ship is drasticall drastically y

experiments on stability stability variation. In this re

reduced reduced when a wave crest moves into the amidship position.

gard, it is is easier to reco record rd many peaks peaks of sta  bili  bility ty vari variat atio ion n than than to obse observ rve e many many caps capsiz izin ing; g;

 Many  Many stud studie ies s on stab stabil ilit ity y in regu regula lar r foll follow ow  ing seas have been made.(1]-[4) Ihey show that

which is why we only measured measured stability variation

 predict:  hydrostatic computations can can predict:  experimental

. in our model experiments. experiments. These results results should should  prov  prove e help helpfu ful l when whenev ever er the the effe effect ctiv ive e wave wave conc concep ept t

values to a certain certain degree. If stabilit stability y varia tion are linear linear phenomena, phenomena, stability stability variation in

is utilized utilized for stability stability criteria.

irregula irregular r seas can easily easily be predicte predicted d with a

2 EXPERIMENTAL TECHNIQUE

linearity. Therefore it is difficult to predict stability in irregular following seas.

In order to evaluate stability stability variation in in : waves, we carried carried out a captive model model tests with with heeled-model heeled-model in both regular waves and in irregu lar waves. The model tested tested was a coastal coastal small small

Grlm[5) proposed proposed the concept concept of an effective effective

trawle trawler^ r^ whose particulars particulars are.shown in Table 1

linear superposed method. However, the relation  betw  betwee een n stab stabil ilit ity y and and wave wave does does not not have have any any

and whose body plan is shewn In Fig.1.

decreased decreased with the increasing increasing Froude number. Both

Fig.2 shows shows a picture picture of the set-up set-up for

the measured values and and the computed computed values show

these experiments. experiments. The model was free with regard regard

that the stabil stability ity is larger larger when a wave trough

only to pitch and heave, heave, which were measured measured by

 moves  moves into into the the cent center er of a model; model; stab stabil ilit ity y is

 pote  potent ntio iome mete ter. r. The heel heel angle angle 9 of 10 degre degrees es

smaller smaller when a wave crest moves into its center.

was

fixed. A rigthing rigthing moment and a ship resist

Fig. 7 shows the the steady steady compo componen nent t 04(0) 04( 0) and

ance acting acting on the heeled heeled model model were measur measured ed by

the second second order varying varying compon component ent G H (2 " ) of

a dynamomete dynamometer. r. The encounter encounter wave heights were

stabilit stability y

recorded recorded by a servo-operate servo-operated d type of wave probe

not smaller than first first order components, stabili

attached attached

the towing carriage. The heeled model

ty in waves is concluded concluded to possess non-linear non-linear

towed with constant constant velocity in following following

 prope  propert rtie ies. s. Furt Furthe her, r, some some simi simila lari rity ty was foun found d

waves generated generated by a flap-type wavemaker with

 betw  betwee een n comp comput uted ed value values s and and meas measur ured ed values values. . On

constant velocity. The irregular irregular waves generated generated

the other hand, there was some discrepancy discrepancy in

in thé basin had a

steady steady component in waves whose height was low.

was

significant significant wave height of

in waves. Since these componen components ts are

about  0.0655 meters and a mean wave period period of

In general, our hydrostatic computations

about 1.39 1.39 seconds seconds in model scale. The spectrum

accounted for the stability stability variation. However,

of waves was defined defined by the ITXC spe spectru ctrum. m. There

the.any the.any dependence on the Froude number cannot be

are some difficulties in the experiments in

interpreted interpreted

irregular irregular following following waves.because a

though the Smith's effect is not small, it too

length of

by such computations.

Moreover,

record record per one run is restrict restricted ed on the length of

cannot explicate speed dependence. depe ndence. We . believe

towing tank. To reach the sufficient sufficient number of

that one must take into account the disturbance

cycles multiple runs were made in irregular

engendered engendered

following following waves.

adequate agreement agreement between empirical empirical experiment experiment

by a ship itself itself in order to obtain

and abstract computation. 3 STABILITY STABILITY IN REGULAR REGULAR POUCHI POU CHING NG SEAS  We comp comput uted ed the the stab stabil ilit ity y var varia iati tion on in regu regu  lar waves by integrating integrating the water pressure

4 STOCHASTIC THEORY ON THE STABILITY

around around the form of the model hull. The disturb

4.1 Grin's effective wave[5]

ances caused by the ship model were assumed to be

 We cons consid ider er that that a shi ship p runs runs in irre irregu gula lar r

negligible as higher order values; the sinkage

  S(o>). following following waves which have the spectrum 

and trim were assumed to be balanced balanced

in the

Grim proposed the concept of the effective waves.

waves. We carried carried out our computat computation ions s in a two

 An irre irregu gula lar r wave wave prof profil ile e arou around nd the ship ship is

fold fold manner, both with the effect effect on the wave orbital velocity ( Smith's effect ) taken into

replaced replaced with a regular regular .wave by a least least square  meth  method od. . The leng length th of the the regu regula lar r wave wave is equa equal l

account and with the same effect neglected.

to the ship length (betwe (between en perpendiculars) L .

results

The crest or the trough of the wave is situate situated d

with the results of experiments described in

at the center of gravity. The stability stability variation variation

section 2. These values were expanded expanded in Fourier

has an apparent peak under this conditio condition n ,which ,which

Beries Beries. . The The phase phase lag ( ( / 2 » ) corresp correspon onds ds to the relative relative position of the model to the wave where

was examined in section 3. Grim called this regu lar wave " the effective wave",of which i eff is

the stability stability is the largest. largest. When the center center of

the amplitude.

Further, we

compared compared our computed computed

gravity of a model is situated situated in any of four  posi  positi tion ons s — the wave wave trou trough gh, , up slop slope, e, cres crest t and and down sl op e—, e— , then its its relative relative position position is 0;0, 0.0-0.25-0.5, 0.0-0.25-0.5, 0.5 and 0.5-0.75-1 0.5-0.75-1.0 .0 respectively. respectively. Figs.3 - 6 show the first-order first-order varying component GM( u> ) of stability stability in regular waves. waves .

The spectrum of the effective wave Seff Seff(o (oi) i) is derived from the least least square method. S.n (u) = S (w) . ( w a L/g) sin (o)e L/2g) , . [ . ;—. .— ] 

 ne -  (u«L/2g) a

(i)

to comp comput utat atio ions ns and and expe experi rime ment nts, s, the the

Furthermore, Grim discussed the stochastic stochastic

amplitude amplitude of response has a peak at the situation ( A /L =1.0 ; A is the wave length length ); roughl roughly y

 prop  proper erti ties es of the the effe effect ctiv ive e wave. wave. Howe Howeve ver r the the

speaking, speaking, the amplitude of stabili stability ty is propor tional to wave amplitude. The values computed computed

stochastic stochastic properties properties of stabilit stability y itself. We derived these properties properties of stabilit stability y as

without the Smith's effect coincided coincided with the

indicated indicated in the following following sections.

 Acco  Accord rdin ing g

values measur measured ed for Fn * 0.0. However, some some dis crepancy crepancy occurred occurred because the measured values

goal which we set out to'achi to'achieve eve was to get the

4.2 4.2 Stability Stability in the effect effective ive wave

One can comp compute ute stab stabili ility ty in the the effect effective ive wave GZw by making making use of the hydr hydrost ostat atic ic method method descri described bed in section section 3. Fig.8 shows shows a comput computa a tional tional result result conc concern erning ing the model ship ship used used in our experiments. This comp computa utatio tion n does not con sider sider the Smith's Smith's effect. effect. The relati relation on betwee between n stabil stability ity and thé effective effective wave are expres expressed sed as follows: GZ»= GZ»=F F (CM)

UM=G

(GZ„)

(2)

This relatio relation n is not only non-lin non-linear ear but also nonnon-me mera raor ory. y. The concept concept of the effective wave can simplify simplify the memory memory effect, effect, that that is, the the freque frequency ncy dependence. dependence. In referr referring ing to double amplitude, amplitude, this. relation relation is also expres expressed sed as follows; OZ, d =f (2Ç.„) = 1 ( C M 4 ) «?•((,„) «?•((,„) + F   '(-CM) (3) To simpli simplify fy the follow following ing formula formula, , we assume assumed d that F( Jeff) is a monotonie monotonie increas increase e function. function.

 pr  probability density function of the stability variation f [.(.is given given by f kl (k, (k, k) =f{j( =f {j(a a (k) (k) , ka (k) (k) )

a (u »

a (k, k) (8)

f |(k is the probab probabili ility ty density density functi function on of the effectiv effective e wave which one can evaluate. evaluate. as the Gaussia Gaussian n process. Therefor Therefore e we can give the average average time  be  between successive zero up-crossing as follows:

 2*1

E [N, (0)]

(0)

 As  As a result, the zero crossing mean period of the stabil stability ity variation variation is equal equal to that that of the effect effective ive wave. 4.5 4.5 The expect expected ed number number of stabil stability ity loss intervals Stabil Stability ity in follow following ing seas is assumed assumed to be G Z (t) =G Zo +G Z„ (t (t)

4.3 4.3 Significan Significant t amplitude amplitude  We can regard the effective wave as a narrow narrow-ba -band nd Gaussian Gaussian process of which the mean is zero. Therefore the double amplitude distribution distribution  p is similar to Rayleigh distribution: ,, n  U., U.,d  CM"8, p e xp t  p (Un") = —  ]  mo 2mo

(4)

where GZ 0 is stabil stability ity in still still water; GZw is stabil stability ity variatio variation n which is treate treated d as a stationa stationary ry random random process. One can regard regard the situa situatio tion n at which GZ is is negati negative ve as the loss of stabili stability. ty. Thus the conditions conditions of stabilit stability y loss intervals intervals are given by

"

Here, mg is the varianc variance e of the effec effectiv tive e wave. Concerning Concerning the relation relation between the effective effective wave and the stabilit stability, y, namely, namely,

 p (GZJ d Q Z , =

the significan significant t double amplitude of the stabili stability ty (GZ>l (GZ>l/3 /3is is proved proved as follows: follows:

'

.

*

.

•

•

•

(11)

Therefore, Therefore, the expected expected number number of stabilit stability y loss loss interv intervals als is given given by

(GZ) (GZ) ,^s=3 Pf PfJUf,") d  C.M  C.M  , (.„« exp [ ] dC.t dC.t. .d    mo :2mo :2mo

'

(GZ. (t) )<0 d t . d (GZ. (t + t) ) > 0 d t'  (t + t) «-GZo GZ» (t) =GZ» =G Z»  (t

(5)

p (Ç.„«) dU«."

(10)

ƒ J

00 f OO fOO poo. -eoJ -e oJ -*o«i -*o«i -O0
(6)

• k k T « (k + ko) 1 ( - k - 0 ) « (k T +ko) •1 (k T -0) f«6i (k, k T, k/k T ) dkdk T d k d k T

-J 0"

4.4 4.4 Zero cros crossin sing g mean period period The number number of up-c up-cros rossin sing g per unit time time of stabil stability ity variation variation is given given by (ko) ] = - P l k l f r f (ko.k) dk E [N,• (k

(7)

2 J -co  -co

where where k is GZw; GZw; k 0 is the thresh threshold old of GZw. The

f° kk T f w« (-ko, -ko, k, kT) d'k'dkt. ° ~°° (12) (1 2)

where k, k T and and k 0 are GZw(t),GZw(t+x) and and GZ Q respectively. f K K K K is the four variable probabilit probability y density density function function of stability. stability. Conside Considering ring the relatio relation n 95 -

 betw  betwee een n stab stabil ilit ity y and and the the effe effect ctiv ive e wave, wave, it is

length length rather than the frequency as a variable variable to

 prov  proved ed that that

carry out our computations computations accurately.

f kk K (k. kT.k, kT) =fj j j j (C, CT, {, CT)

4.6 Probability of capsizing  Acco  Accord rdin ing g to Krap Krappi ping nger er [7] [7] and and Hira Hiraya yama ma

3 (î. ÎT, C. CT)

[11 J,

' a  (k, kT. k. kT) - ' « j j w c w . a ( k T ) .

the probabil probability ity of capsizing capsizing is defined as

the probability probability that a ship will have capsized in a stationary random sea before time tc. He

ka (k) , ka (kT) , (a (k) à (kr) ) » (is)

assumed assumed

f/ / f / is the four variable probability probability density

scribed scribed by a Poisson process. Further, the case

function of the effective effective wave. Since the effec

that stability stability loss interval exceeds the time

tive wa"e is regarded regarded as a zero mean mea n valued

interval T c is regarded regarded as capsizing. Therefore Therefore

Gaussian Gaussian process, one can give f/ r } V by

the expected capsizing rate v is given given by

that an act of capsizing capsizing could be de

'= f

E tNk (t) ] dt

(1 5)

't.

^.4,Mài^'"fc[riTiuc''

+ b î 

+ dCCT  + 1 / 2 e  U'+C t ' ) ) ]

 More  Moreov over er one can deri derive ve the the prob probab abil ilit ity y capsizing capsizing as follows:

° <<-<*>

(14)

P (t (te) = l - e x p (-v to) 

of

(16)

where the coefficients are: 5 STABILITY STABILITY IN IRREGU IRREGULAR LAR FOLLOWING FOLLOWING SEAS

a= (m a -m 8  (t) ) [ (m 0 -m 0  (t) ) • ( ma + me ( 0 ) - m i ' ( t) ] b=m i (r) (r ) [ (mo - m 0  (t) ) • (nn+ma (t) ) -mi 8  (t) ]

The

 mo'—  mo '—mo mo ' (t)

ƒ

The significant significant amplitude of stability stability ivYH/313 shown shown in in Fig.9. The (Hi/3)' = (GZ)1/3/( ivYH/313  meas  measur ured ed valu values es decr decrea ease se with with the the incr increa easi sing ng

o

Froude numb number. er.. . This tendency tendency is similar similar to that that in regular regular waves. waves . The measured values values for Fn=0.0 Fn=0.0

>0

 Mm.«  Mm.«

prediction prediction de d e

scribed scribed in section 4.

8 e=m« {mo —mo ' (t) ) — mo mi 8 (t)

I AI

their experimental experimental

results with the computationa computational l

d = — m s (t) (mo 1- mo' (O ) +m 0 (t) mi" (t) e'-d'

authors compared compared

are much larger larger than the computed computed values . The component component wave length length corresponding corresponding to the criti

•

cal freguenc freguency y O =u «^/g =1/4 =1/4 where Fn=0.2 Fn=0.2 is

S a n ((de) d&)« '

equal equal to the ship length; length; U is the ship speed. speed. In this case, the effective effective wave concept gives us a

««'Sali (»a) d Ua

good approximation, approximation, because the encounter encounter spec trum trum has a recognizable recognizable peak where O =1/4.  The  meas  measur ured ed value values s for for Fn=0 Fn=0.2 .2 coin coinci cide des s with with the the

. m o (t) = 1 :

S.fl S.fl («Ja) O 6 S Wa t d   <•>.

computed computed values. However, hydrostatic computat computation ion in regular waves does not accurately accurately estimate estimate

. " O

 mi (T ) = - | •

w .S .S .f .f . ( w j s i n u . t d u ,

 meas  measur ured ed value values s at such such a spe speed. ed. . Thus we conjec conjec tured tured that any errors of the hydrostatic computa computa

»

^ f Wo.«  m« (t ) = I ««'Salt (tl«) o O S U« T d u . J  0

tion, or any errors inherent inherent in the effective effective

If «"max «"max take3 the value of infinity, our computation computation involves case which have more than

wave concept ,or ,or any errors in our experiment experiment cancelled cancelled one another out.

two peaks in a concerned concerned interval. interval. Some conven

In Fig.10, the zero up-crossing up-crossing mean periods T z J i 7 E are shown. shown. At the high speed speed situat situation ion

tional methods have been presented presented to avoid avoid such  prob  proble lem m as this this.[ .[10 101 1 How Howev ever er, , we adop adopt t anot anothe her r

discrepancy between the measured Fn>0.2, some discrepancy values values and 'calcu 'calculat lated ed values occurred.

 meth  method od, , in which which «" max is 2 n a  /t (1/2<« <1  In  <1 ). ) . In this method, method, the th e periodic behavio behaviour ur of the compo

This

seemed seemed to be caused caused by the fact that the analyz analyzed ed

nent waves does do es not contribute contribute to the final Further the final value did not not depend on value. Further

time history was actually actually the sum of shor short t time histories measured measured in each separat separated ed experiment.

the arbitrary arbitrary value o f « with thé restricted restricted

Fig.11 shows the expected number of sta  bili  bility ty loss loss inte interv rval als. s. Acco Accord rdin ing g to to the the meas measur ured ed

range (1/2< a <1). <1) . In additio addition, n, we used the wave-

- 96

values and the computed values, the long stabili ty loss intervals occur at high speed speed situations. The computed values are smaller than the measured tendency is evident values. In particular, this tendency in a high speed situation. To obtain a satisfac tory agreement, we must satisfactorily predict amplitude, time period, period, mean mean value and so on. He also carried out computation computationss in which the ex  periment  perimental al mean values correct corrected ed the thr thresh eshold old of stability. The dotted lines in Fig.11 show these results, which are much larger than than the original original computational results. Hydrostatic computation in regular waves does not well predict the steady steady component of stability. This component possibly involves involves higher order hydrodynamic hydrodynamic forces. There fore, in order to predict predict the expected expected number of stability loss intervals, we must correctly com  pute the mean value in in irre irregul gular ar waves. Fig.12 Fig.1 2 shows sh ows the computed probability of capsizing. capsizing . In these computations, we used the computed computed values of E[N E[N]] with the correction men tioned above. We assumed that the initial initial sta  bility  bility GM -/ Bw as 0.338; 0.338; the time of capsi capsizin zingg T c was 0.4. This diagram shows that the probability of capsizing is Strongly Strongly dependent on the Froude number. In general, genera l, the results of the stochastic  predictio  predictionn based on the effective effective wave con concep ceptt are similar to the experimental results. However, we cannot conclude that this method can be used for a practical practical purpose as long as the authors discussed. 6 CONCLUSION The authors believed they have confirmed that hydrostatic computati computation' on' can to some extent  predict  predict sta stabili bility ty variatio variationn measured measured in in regular regular following seas, pur stochastic prediction method  made use of hydrosta hydrostatic tic com computa putatio tionn and and the ef fective wave concept, and was verified verified by the stability measurement of a model in irregular following waves. .  ACKNOWLE  ACKNOWLEDGEM DGEMENTS ENTS  We are grateful grateful to Mr. Mr.S.S S.Suzuk uzukii of Nat Nationa ionall Research Institute of Fisheries Engineering for his help in model experiments. experiments. We would would like like to thank Mr. W.Hansen, W.Hans en, CE2 port lecturer,for his • ' syntactic advice. advi ce. The computations were carried out by ACOS850 at the Computing Center for Research in Agriculture, Forestry and Fisheries. REFERENCES 1. Watanaba,y., "On'the Dynamic•Properties of the Transverse Instability; Instability; of a ship due to.

Pitching," Pitching," Journal Journal of the Society of Naval Archi tects of Japan (J.S.N.A. ),  Vol. 53, 1934. 2. Paulling,J.R., "The Transverse Stability of a Ship in a Longitudinal Seâway,"  J.S.R.,  Vol.4,No  Vol.4,No.4,19 .4,1961. 61. 3. Upahl.E., U pahl.E., "Ermittlung der Schiffsstabilitat Schiffsstabilitat . von Hec Hecktr ktrawl awlem em in regelmässi regelmässigem gem längslaufend längslaufendem em Seegang," Schiffbauforschung, 18,3/4,1979. 4. Hamamotó M. and Nomoto,K., Nomoto,K., "Transvers "Transversee Sta  bility  bility of Ships Ships in a Follo Followin wingg Sea," Pro Procee ceedin dingg of 2nd International Conference on Stability of ships and Ocean Vehicles (STAB'82),1982. 5. Grim,0., "Beitrag "Beitrag zu dem Problem Problem der Sicher Sicher heit des Schiffes im Seegang," Schiff und Hafen, Heft 6,1961. 6. Krappinger,6., "über Kenterkriterion,"Schiffstechnik,Heft 48,1962. Kas tner,S., "Hebelkurven "Hebelkurven in unregelmassigem 7.  7.  Kastner,S., Seegang," Schiffstechnik,Bd Schiffstechnik,Bd 17,Heft 88,1970. 88,19 70. 8. Helas,G., "intact "intact Stability of Ships in Following Following Waves," Wav es," STAB'82,1982. 9. Price,W.G. and Bishop,R.E.D., "Probabilistic Theory of Ship Dynamics," Dyn amics," Chapman Chapman and Hall,1974. 10.  Takeuchi,S. and Yamamoto,Y., "Approximate Distribution and Simulation of Successive 'Ex tremes for Gaussian Random Process,"  J.S.N.A.,  Vol.131,19  Vol.131,1972. 72. "Experimental Study on the 11. Hirayama,T., "Experimental Probability of Capsizing of a Fishing Vessel in Beam Irregular Waves," J.S.N.A.,Vol.154,1983.

0.60 Qv£w 0.40

Table 1 principal particul ars of ship Length B.P. Breadth ..Draft Fore

Lpp[m] 8 [m] df[m] da[m] V [m3] 1cb[m]

 Aft Displacement  Volume C.6. from mfc ship

Ship 14.40 3.05 0.35 1.396 27.56 1.28

 Model 2.25 0.477 0.055 0.2181 0.1051 0.201

FN = 0.1   Measured

n

0.20 0.00

1.0

0.0

2.0

3.0

A/L

1.25

4.0

5.0

Fn =0.1

f/2*

1.00

0.75

rW

h 

0.50 0.25 0.0

AP IS

a 

 

Measured

computed without Smith' s effect computed with Smi th' s effect 2.0

\.0

3.0

4.0

5.0

Fig.4 Stability variation in regular waves

Fig.1 Lines of ship

0.60 GM/£ W

FN =• 0.3   Measured

13

0.40 0.20 3.0  4.0  A/L

0.00 0.0 1.25 e/2n

m

H

 m

1-

5.0

Fn = 0.3

 m

1.00 0.75 Fig.2 PhoLoyrdph of experiment in irregular waves

0.60 GrW,w

0.50 0.250.0

1.0

3.0 4.0 5.0  A/L Fig.5 Stability variation in regular waves

01

0.30

  Measured

0.20

0.20

0.10 1.0

2.0

3.0 A/L.

1.25

4,0

0.50 0.25 0.0

Fn = 0.0

f • .. 'D- Measured —- computed withou t Smith' s ef fe ct --- - computed with Smith 's ef fe ct 2.0

3.0

A/L

4.0

13

0.0

1.00  t/2n

J^Gffl ° ". 01

1.0

0.00

5.0

1.00 0.73

2.0

0.40

FN = 0.0

0.0

a

  Measured computed without Smith's effect computed with Smith's effect

0.40

0.00

l,

5.0

0.50 0.00

0.2

"ill

DD

0.4

o

  Measured 0.6 0.8 1.0x10 H/A

m o

• Comput»d with out Smith '» «f f» ot - - - - Computed with Smit h'*   »ttmet D

0. 0

0. 2

0. 4

  Measured 0.6 0. 0 1.0x1 0

H/A

Fxg.6 St ab il it y var iat ipn in regular waves

Fig.3 Stability variation in regular waves

- 98 -

Computed ult hou t Smit h's Computed ul th Smit h's

1.60

1

FN = 0.1 À/L  = 1.0

GMlî! OMR

1.20

ef fe ot ef fe ct

0.20 EIN]

1—

Measured

FN = 0 .0

Computed Computed by using exp. mean value

0.15

0.80 0.10

0.40

/ "

0.00 -0.40

 \

0.05

torn

v.

m

0.0

0.B0 SHÈ-i 0.60

0.2

  Measured 0.6 0.B 1.0x10 H/A

0.4

-(

FN = 0.1 ÀA.  = 1.0

1

0.00 0.0

0.2

0.B

1.0

1.2

t

0.20 EIN]

-

ÜL0.6

0.4

1.4 1SEC)

Measured FN = 0.1 Computed Computed by using exp. mean value

0.15

0.40 0.20


" 13

0.00

0. 0

1

0. 2

  Measured 1 1 0. 6 0.8 1.0x10

1

0.4

Fig.7 Higher order components of

xl0?n] 1

 M

0.05

0.00 0.0

stability variation 0.60 GZw 0.40

0.10

0.2

0.20 EIN]

1

0.4

rm , 0.6 0.8

t

1.0

1.2

-i 

1-

1.4

(SEC)

Me as ur ed FN = 0.2 Computed Computed by using exp. mean value

0.15

0.20 0.10

0.00

0 05

-0.20 -0.40

0 00

1

-0.6

-0.3

0.0.

»eff

0.3

0.6x10

M

Fig.8 Computed stability

0.2

0.4

0.6

0.0

t

1.0

1.2

1.4

tSEC]

Measured FN = '0.3 Computed Computed by using exp. mean value

EIN] 0. 15

13

  Measured — Computed

0.60

0.10

0.40

0.05

0.20 0.00

0.0

0. 20

in effective wave (model scale)

'xlB - 1 0.80

•4T-47n.l^|4+£P.[ , r-, r-.

0.00 

0.0

0.1

0.2

0.3

0.4

KT-,.

0.0

0.2

0.4

toy ra^q..hi._r >. fl-J-i 0.6 0.8 1.0 1.2 1.4 t

FN Fig.9 Sign ificant amplitude of stability variation

CSECJ

Fig.11 Expected numbers of stability loss intervals (model scale; GMo/B=0.338)

10.00 TM/S7C'

1.00 P 0.75

8.00

/»'»it - •  _./- - * . : • ;•-*•—"'*^*FN° B.a

Computed   Measured

n

0.2

0.3

FN Fig.10 Zero crossing periods of stability variation

0.0

0.4

1.0

2.0

3.0

to 

4.0

5.0 (min);

Fig.12 Computed probability of capsizing (model scale; GMo/B=0.338;Tc=0.4 sec)

- 99 -

H.Us»da graduated trois Osaka University, Japan,la 1980, He« bs trorfts at Rational Research lo st I ta ta of Fis her ies Engine ering, Japan. Re has Been •D ia le d 1P researches on the saf ety of fi sh log boats f er seven years. .••. 'Y. Tasakosb 1 graduated fr on the Yokohama Rational University, Japan,in 1972. He bas been a cfelef of A l p Performance Seetlon at Nati onal Rosoarcb Ins t1 tot e of Fis her ies Enginee ring,si nce '1984, He Is interested In sbip motion and stabilty of flablng boats.

Third International Conference on Stability of Ships and Ocean Vehicles, Gdarisk, Sept. 1986

wmw> Paper 2.3

 MODEL EXPERIMENTS ON INCLINED SHIP IN WAVES Cao,

Zhen-Hai; Li, Jun-Xing

 ABSTRACT The characteristics of an Inclined iship in waves is an important aspect of [investigation from the point of view of [stability and safety. The angle of incli nation of a ship can arise from the stea dy bean wind, shifts of cargo and partly flooding. A project of model experiments jon inclined ship in waves has been carri ed out in the seakeeping model basin of OSSRC. The ship form under test is a wide 'shallow hull form. All of these model ex  periments were carried out at zero speed In regular waves.Various angles of incu  bation of the model in its mean position land various vertical heights of center of gravity of the model were adopted in the experiments. The comparison with the ex  perimental results of the up-right condi tion of the model in regular beam sea is £leo included,and the detailed discussion  bf the investigation is made., jl. INTRODUCTION ; The stability of a ship is a prior aspect to be considered by Naval Archite ct in the design stage. The designers al ways make their new design have adequate restoring moment to ensure the safety in navigation. Besides,international classi fication societies and govermental regu latory bodies of maritime nations have their own stringent requirements and re gulations. They all stipulate the minimum Stability critera for various category of ships. Neverthiess, despite the precau tionary measure present in the ship de sign process, loss of life and ships due i

to capsizing

is

still

occuring. On the

 basis of statistic investigations of ac cidents in the  past, the causes of capsiz4ng of various category of ships are very

different. The most capsizing accidents of cargo ship are due to shifts of cargo /1/. According to the amount and bias of cargo to be shift,the ship capsizes imme diately or hàs a steady angle of'inclina tion. The inclination angle of a ship can also arise from the presence of the stea dy beam wind,partly flooding, shipping of water or ice trapped on deck, etc. If a ship in inclined condition has an asymme tric athwartship underwater geometry, the hydrodynamic properties and the characte ristics of motion response should be di fferent from the up-right condition. The coupling effects of six modes of inclined ship motions in waves are more complex than the usual case in upright condition. To investigate the characteristics of an Inclined ship in.waves is an impor tant aspect for ship safety and surviv ability of a damged vessel. A few authors have developed theoretical methods to  predict the characteristics of inclined ship in waves/2//3/. They used the source sink distribution technique to compute (the hydrodynamic coefficients of two di  mensional, asymmetrical underwater cross section and strip theory to evaluate the five modes of motion except the surge. Prom their study it was revealed that a ship in its inclined condition can be ex cited to higher roll motion than the up right condition under certain circumstan Undoubtedly, it is an unfavourable ces. jfactor for stability of inclined ship if It has higher roll response in waves. This paper describes the results of  model experiments on inclined ship In re gular waves which were carried out in the seakeeping model basin of CSSRC. The ship form under test is a wide shallow hull fcith two different vertical heights of 101 -

C G . and two different inclination angles. Por the sake of comparison, the model in up-right condition is also tested. The  present paper describes the model experi  ment rerults, and some relevant discu ssions is also given. 2.MODEL BCPBRIMKHTS f The ship form under test is a wide fshallow hull with the parallel middle body -extending to about fourty percent of the ship length. Ho appendages were attached except the bilge keels. The principal'^characteristics of the model are presented ; in Table 1 and the body plan is shown in jPig.1. i The principal dimensions of the iseakeeping basin in which the model expé rimenta were carried out is 69 meters in 'length, 46 meters in breadth and 4 meters, |in depth. Pneumatic type wave-maker is equipped. All of these model experiments, 'were carried out at zero speed in regular iwaves without restraint on any mode of  motion. The measure of three translations and three angular displacements were pick ed up by six degree of freedom instrument whioh was developed by CSSRC. The wave (parameters were measured by capacity type

wave probe. The up-right condition has two 'different vertical heights of C.G. as shown in Tablel.By translating the ballast jiron to the portboard in these two heights :Of O.G.,we could make the model reach the  predetermined inclination angles,  i.e.^«5j degree,^-10.degree,to.simulate the shifts: of cargo. Thus, the inclined model has lower freeboard at port Bide and higher' freeboard at starboard side. The displace-;  ments of the model in inclined condition and in up-right condition are the same. We: specify here that the heading angle of incident wave is 180 degree for head wave; 90 degree for beam wave which incidence is. from higher freeboard to lower freeboard. 'The wave length was varied from 1.4 m to 6.5 m and the wave height was approximate-! ly kept in the range from 50 mm to 60 mm.' Before the experiments were carried out in waves, the roll decay tests for various situation were carried out in calm water' i to measure the roll damping coefficients and roll periods of the model with various heights of C O . and various angles of inclination. The immersed midship cross .sections at $ « 5 degree and  «4 »10 degree itogether with the locations of the center of buoyancy and gravity are shown in

Table 1 Principal Characteristics;of Model in up-right condition

5

Length, L Breadth at midship, B Draft, T

i

t

2i80 m 0.512 m O+096 m

5J33

B/T Displacement Volume Block Coefficient, G b L.CB. L.CG.  Vertical height of C.B.  Vertical height of C.G. Pitch radius of gyration Transverse radius of gyrati  Metacentric height, GM

i

I i

; '•I I

!

+ I

, • ä

0I1O8 m  Ol 783 oLo915 m fore of 0.0915 m fore of olo496 m 0j107 m; 0.188 m 0I28 L 0i34B oLl7 m; 0.09 m

Pig. 1 Body Plan

- 102 -

10th station 10th station above the keel above the keel ' ; •

Flg. 2 Midship sections at various inclination angles

Pig

Righting moment arm versus inclination angle

Pig.2.The Btatic stability oharateristics \ of the model with two different vertical heights of C O . is present in Fig.3 by the curve of righting moment arm GZ versus heel angle <£.  All of the model experiment results were described in the form of nondimensional amplitude  Z/I/5V»$/MA* etc, and hon-dimenaional wave length -VL. ! As far as the safety of inclined ship in waves is concered, the roll and heave  motions always play an important role. Besides, the roll and heave are the main  modes of motion of a ship in beam seas , so jthey are the only topics we shall discuss. Of course, the pitch of inclined model should be taken into account in head seas. t

•The responses of roll and heave against the non-dimensional wave length */ L are described in Fig.4 to Pig.7. The roll  periods in various situations obtained from deoay test in calm water are shown ;kn:Table 2. Table 2 The roll periods of the model, in various situation 0 degree •

5 degree

10 degree

^

0.107m 0.188m

1.014 sec 0.932 sec 1.69 seo 1.62 seo

0.923 sec 1.57 sec

3.RESULTS AMD DISCUSSIONS Two different vertical heights of C S .  of the model under tests are all above the water line. The ratios of KG/T are 1.11 and 1.96 respectively. For thé condition of lower vertical height of C G . , KG-0.107m, the peak value of roll response is lower in up-right condition and nearly no peak exists in inclined con dition at$«10 degree. The roll responses are nearly the same in inclined condition for various wave lengthes in the range of experiment. For inclined condition at  
-i 103 -

 faS", 4'Stf  - "-

fats'; *>sS- 

* T-

ft

to

*-S*-X=*Z=i.-=M = r.--.

r?


ts

to

to

ts

y L

tO

06

to

ts

ts

%

[

 ft

?0

u

to *-».*—* 

06

IV

•

 fan'; A'tf —•—

%

t.~
•_ (6

&

&

%

05

Pig.4 Roll and.heave responses for  lower; vertical height of C G . (KG«0.107m)  in]  bean waves at  up-right condition  and inclined condition ($«5 degree)

a

&

•

%

Pig.5 Soil and heave responses for  lower'  Vertical height of C O . (KG-0.107m) in beam waves at  inclined condition  (£-10 degree)

fag;   -a»*' — fag)  ü-fflf  --  k°0'i
% IS

ts

te

% IS

tO ^

-»- —ta.= _

*Ü

to

06 OS

to

to

ts

to

ts

%

05

i-fi-niftr——

 — i"ój4yf 

—

(0

ts

 A

4o

%

4:V .

A fats',-«•&-  il

 30  30

% .

 to

M)

to

.".-••',  A-to'j-4°9o'—~—

 40

&

 to

 to  .06

to

ts

to

&

ty L

05

 pig.6 Roll and heave responses for  hjgher vertical height of C.G.(KG-0.188m) in beam waves at up-right condition and    inclined) condition (£-5 degree)

to

fS

to

»

y.

Pig.7 Roll and heave responses for  higher! vertical height of C.G.  (KG-0.188m) in  beam waves at  Inclined condition (£«10| degree) *'•

- 104 -

are different and the range of significant response is wider in inclined condition than in up-right condition, that is to say that the inclined ship can be excited easily by waves of various lengthes. There is no remarkable difference of peak value ,  between the up-right condition and the in clined condition at <£-5 degree. But for inclined condition at 4>. -10 degree, the'  peak value of roll response for wave head-: ing/i »90 degree is higher than that of » . . . roll response for wave heading ja =270 de gree. The heave response for inclined  model is higher than for up-right model.! This is of the same tendency as in the' condition of lower vertical height of C G . But there is an evident characteristics existing in inclined model, i.e. the loca tion of the peak of the heave response in abscissa is close to the peak of roll response. This phenomenon does not exist in üp-right condition.From the cómparision óf heave responses between the conditions |of two different vertical heights of C O . , it is observed that the difference is not evident between two up-right conditions.: The heave response for the higher vertical height of C O . is obviously higher than that for the lower vertical height of C.6. in Inclined condition. All of these pheno  mena are due to the coupling effect  between heave and roll in incline^ condition. ! For a symmetric up-right model in regular head waves, the occurrence of roll  motion would not be expected. However, i the model in inclined condition, the roll1  motion would occur in head waves because) the model no longer has the symmetric hydrostatic and hydrodynamic properties./ .The experimental results of an inclined:  model In head waves with higher vertical, height of C.G. at inclination angle %° 1 0 degree are shown in Pig.8.Prom this figure jlt can be observed that the location 'oftV the peak of roll response in abscissa is close to those of peaks of heave andj ipitch. That is to say the coupling effect!  between vertical and horizontal plane mo tions of an inclined ship is important. 4.C0NCLUDIgQ REMARKS / Prom the model experiments, the; following findings could be made« ; 1. The différence in roll response  between inclined condition and up-right {condition deponds on both of the vertical -10.5.

height of C O . and the angle of' inclina-" tion. Oenerally speaking, the peak value of roll response óf inclined model is not higher than that of up-right condition. But the range of significant response nearby the peak of roll response óf the inclined model is wider than that of the up-right model. 2. Under certain circumstance the roll responses of inclined model at wave he adin gs » 90 degree could be higher than those at wave heading /t-270 degree. 3. The heave responses of inclined  model are higher than those of üp-rlght  model. Particularly, there is a  peak of. heave response nearby the peak of roll response in inclined condition. 4. The inclined model can be excited to à large roll motion by head waves. Under this circumstance the peak of roll response at the location in abscissa is in correspondance with the peaks of heave and  pitch.

%

05

./' 06

t-0

to

&

-15

y L

-_.— non —.__

Pitch

Ato •

'

 /*\

"

w

 \  \

 M

^~2"v

-.• •

06

t-0

15

M

& •

%

Fig.8 Roll, pitch and heave responses for iigher vertical height of 0.0. (KG=0.188m) in head waves at inclined condition (#-10 degree)   •:.•••

iJOMENCLATURE •'.

I»

.B T

ok **

 z* ft.
<& T* KG K

x

OM

 A

Ship model length Breadth at midship of thé  model Draft of the model Block coefficient  Wave amplitude Heave amplitude Pitch amplitude Roll amplitude Initial angle of inclination Roll period  Vertical height of CO.above the keel  Wave number  (-271/A)  Wave length  Metacentric height  Wave heading anglet head wave (,U -180 degree ), and  beam wave from higher free  board to lower freeboard (yU-90 degree)

•RSFEBBNCSS | 1. Takaishi.T.,"consideration on thé iDangerous Situations leading to capsiee of Ships in Waves", Proceedings of Stability ;*82. r 2. Lee, C.M. et al, "Prediction of  motion of Ships in Damaged condition in  Waves",  Proceedings of Stability '82. 3. Kobayashi.M.."Hydrodynamio forces ,and moments acting on two-dimensional iasymmetrical bodies" Proceedings of Stabil 'lity '75.

Cao, Zhen-Hai Graduated from Dept. of N.A..Shanghai Jiao-Tung University in 1955. He has been working for China Ship Scientific Research] Center as a research Engineer since then. j In the past,he worked in the field of ship:  performance, seakeeplng and model testing' technique, including the design and super-; vision of the construction of seakeeplng;  basin. He was a member of the committee; for revising the rules on Stability fori jthe China Classification Society in 1973* j \ In recent years he has been working I in the field of stability of ships,and was; 'invited to be a committee member of the > Stability and Load Line Technical Sub- j Committee of Register of Shipping of the! People's Republio of China. j

Li, Jun-Xing ; Graduated from Dept. of H.A., south China Engineering Institute in 1964. He has served China Ship Scientific Reaearoh Center as a research engineer ever since. He has been working in the field of seakeeping and performance of high speed vessels.

106. -

Third International Conference on Stability of Ships and Ocean Vehicle^   Gdarisk tSej>t.  1986  

*"l\ Paper 2.4

MODEL EXPERIMENTS ON  CAPSIZING OF A  LARGE STERN TRAWLER

M.  Kan, T.  Seruta, T. Okuyama

ABSTRACT

Results are presented of model tests performed to Investigate into the accident of a large stern trawler, which capsized and fouiriered at the Bering Sea with 32 victims except only, one survivor. Model .experiments on ship motions in her various intact and flooded conditions did not show any evidence of poor seakeeping qualities . Measurements of th e in ta ct st a b i l i t y showed a good agre emen t w it h t he ca lc ul at io n which niet the criteria with a substantial margin. Stabilities with the free water in the'factory and in process of flooding through the garbage shoots were also measured, and besides capsizing tests of the initially heeled model with open garbage shoots were performed. Prom these results and other examinations including the testimony of the survivor, it was concluded that the capsize had been caused by water flooding through the unclosed garbage shoots into the factory and then probably into the fish hold. 1. 

INTRODUCTION On 6th January 1982 at about 16:00, a large ste rn trawl er of 549 gross tonnage, sim il ar t o th e GAUL (1 J, cap sized and foundered during ha uling ne t in the Bering Sea at a position 54°05' North 178° 25 ' West, 140 mil es n ort h of the Tanaga Isl and i n Aleutian Islands (Fig.1). Out of 33 crews on board, 32 members lo st t he ir li ve s and only th ir d off ice r

was saved by her friend ship. The wind speed was reported as about 15 m/sec from the east, and the significant wave height as about 5 m, which was consistent with the estimation by the Meteorological Agency. The wave period was a l s o es ti ma te d a s abou t 9 seco nds. From the wind speed of 15 m/sec and the ai r temperat ure of abo ut 1 °c, t he po s si bi l it y of icin g on the ship was considered very little. She had two decks between which there was a large fish processing factory and there were two openings called garbage shoots with the size of 50cm x 40cm on her sta rbo ard s i d e , and on e on the port side in the factory (Fig.2). There was no testimony concerning whether th os e garb age sho ots were clo sed o r l e f t open. It was reported th at i t had taken about 17 min ute s from be gin nin g of th e heel to the capsize. The Min ist ry of Tra nsp ort or gan ize d th e committee to examine the cause of the accident and to study the saf et y of a fi sh in g sh ip in an o p er a ti on con di ti on . The model experiments de sc ri be d her e were performed as a work of t h e committee I2].

ip

Fig.1

Spot óf Accident and Weather Chart

lb

Î0

30

40
Fig.2 General Arrangement

- 10 7 -

3cm, the inner breadth or volume of the factory space and other compartments was decreased to th at extent. Though the sheer of deck was similar to the ship , the camber was neg lec ted. In order to avo id flooding into the inside when the model was capsized, or to avoid the leak of the water in some compartment to another one, the openings on the deck had the water tight lids using rubber packings and sp ri ng c l a s p s . The li d s were made by the transparent materials to observe the ins id e of the model. The main superstructures above the upper deck were also made as similar as possible in order to examine the effects of the wind. Physical values described afte r here are giv en in fu l l sc al e according to the Froude's similarity law.

2. Model The principal items arri the load condition of the model are shown in Table 1. As a result of an inquiry of the committee, the load condition of the ship at the  time of an accident was estimated as shown in the tabl e.T he drafts of the model are slightly different from the corresponding values of the estimation, perhaps due to the problem   of the accuracy of the model and/or  the  accuracy of. the hydrostatic calculations of the ship.

Table 1 Principal Items ITEMS DISPLACEMENT (I on) LENGTH LPp ( m ) BREADTH 8 ( m ) DRAFT df  (mf d0 ( m ) d m  (m) TRIM BY STERN (  m ) KG (m ) GM ( m )

MODEL 0.172 2.50 0.531 0.104 (0.097)** 0.318 (0.318) 0.211 (0.2 07) 0.214 (0.221) 0.231 0.0394

SHIP 1477 50.8 10.8 2.11 6.46 4.29 4.35 4.69 0.80 « drolls are measu red from bottom of box keel •«» values In brackets are estimated by the committee

3. Rolling in Beam Sea Model te st s on the roll ing cha rac ter isti cs in beam seas were carried out to examine the effects of GM, flooded water, heel and wave steepness. Fig.5 shows the effect of GM. Since the natural rolling period of the model" was confirmed to be reasonable by comparison with the full; scale te st of the si st er sh ip "No.27 Akebono Maru", i t could be said that GM  less than 0.8 m  had an effect of decreasing the rolling amplitude for the presumed wave length X/L=2.5.

=£tii==Ft

MARKS GM(rn)

-9" -O-  -O- 

-*- 

5.0 h

1.0 0.« 0.« 0.4

UPRIGHT

WITHOUT FREE WATER 1: CORRESPONDING TO ROLLING PERIOD

A = '/30

• 4.0

1.0 2.0 3.0 4.0 SO 60   V L ; Photo .1

View o l  Model.

Fig.5 Effect of  GM on Roll

BUJ.WAiîKjOP

% 5.0

D.LW.L.

F ig .4

WITH FREE WATER IN FACTORV «:   WEIGHT  OFF. W: MARKS «ID GM-0.8m (01 K»0Ion) 0 25 -O—a— SO -o— 100

Body Plat»

Fi g. 3 shown the arra ngeme nt of th e mode), and Photo. 1  shov.-s i t s . view. The boeiy plan i s shown in Fi g. 4. To'examine the ef fe ct s of t he fl ood ! rig water in 1'i.o fa ct ory, th e compartment of the fa ct or y, the hjfx'h in the fac tory and th re e gar ba ge sh oo ts lanre made as s im ila r, to the ship as po ss ib le . However,"since the thlcknuss of the model was about

.' .;

UPRIGHT

"0

1.0 2.0 3.0 4.0 5.0 6.0

Vl

Fig.6 Effect of Flooded Water on Roll ':

- '108

Fig.6 shows the effects of the'flooded water in the factory. For •  A'/L=»2.5  the free water

decreases the rolling response. Fig.7 is another example of the effects of the free water and GM. Effects of the heel is shown in Fig.8. Since the heel of the ship has an effect of shifting the natural rolling period  to the shorter range, the rolling amplitude increases for V L = 2 . 5 . But this  may not be the case of the final situation of the accident because of the possible existence of the

methods and ca lcu lat ion shows a su ff ic ie nt accuracy of the experiment as well as the calculation program of the int ac t st ab il it y.

COUNTER I.-.WHSHT.

flooded water. It can be considered that the flooded water in the factory even in the listed condition will decrease the roll response, which is consistent with the survivor's testimony that there was little roll motion after listing to the starboard. In the same figure the effect of the wave steepness is shown as expected.

0/ '9* 5.0

MARKS GM m) UPRIGHT Ie " ^ 11.0 -° 1 of=25 ton |o o | 0.o.e  | — « z o

e

•

0.

o.

Fig.9 Apparatus for Measurement of Stability

GM« U> m

4.0

 f~\

 I 

!

0.8 m \ 

0.6 m

 WITHOUT FREE WATER STARBOARD

»

3.0 2.0 . 1.0 "0

1.0 2.0 3.0 4.0 5.0 6.0

VL

Fig. 7  Effect of Flooded water and GM on Roll

o EXP. BY PRESENT METHOD • EXP. BY WEIGHT SHIFT .

 — CAL. i & V«K> M«-tot HEELANGTE 20*10 LEE  SIDE -o20"TO WEATHER SIDE  —A ~ UPRIGHT GM°08mAT  UPRIGHT CONDITION

-•-*—

10  20 30 «0 50 60 70 e (deg.) Fig.10 Intact Stability a :  WEIGHT Of FREE VWTER IN FACTORY ( ton )

Fi g. 8

1.0 2.0 3.0 4 0 5.0 6.0 VL • Effec t of Heel and Wave Stee pness on Roll

These results on the roll motion in various conditions do not show any evidence of poor characteristics connected with the capsize.

4.   Measurement of Stability St at ic st ab il it y was measured in v ari ous conditions. Fig.9 shows an apparatus for measuring the rig hti ng moment under the trim f ree co nd it io n. Measured moment was corrected by subtracting the mcment act ing on the load c e l l a f t e r removing t he model. Fi g.1 0 i s an example of the measured int ac t stability by using the above apparatus and by means of th e weight s h i f t . Good agreement among both

20  30 40 50 60 70 O (deg.) Fig.11

Stability with Free Water in Factory

Fig.11 is an example of themeasured righting arm   of the flooded condition. Since there is not a complete similarity of the factory between the  model and the ship, quantitative agreement with the calculation is not so good. An example of the -•409 -

calculation of the flooded condition which was

carried    out. Fig. 14 shows an example in still

 performed for the actual ship, is shown in Pig.12.

water, which shows the effects of GM, number or the

The righting moments in case of the open

opening, and the initial heel angle on the elapse

garbage shoots and the opened   or closed hatch in

time to the capsize and the final bal lance angle

the factory were measured as shown in Fig. 13. The

of the model in case of the closed hatch in the

heel angle of about 9=13 ° corresponds to one of the

factory'. As expected from   the mentioned abo ve,

 beginning of the flooding through the garbage

there exist some cases where the capsize does not

shoot, which are not exactly similar to the actual

occur even if the flooding is allowed. Photo.2 is

ship because of the thickness of the model. At

an example of a seriese of shot.

about 0=24 ° the water accumulated in the factory  begins to spill into the fish hold through the hatch .  — - J S N D n

nciamini* MSP.

(lonl

do(rn) •Mm) 8 *  TOM  fm> MFUMAMGU

X.GM=Q6m T—*-»—  o—

s A B c 32 92 a 1*8219 !S91!S I9M.19 19* 1.98 199 6.«» 9.67 697 il) «.n *.W -on -an -1.18 «so «« «.71

n

Si.s •   FBB: WHERtFl OWlNOFISH

«0

30

Ü

ae as

GZ (m)

3 100f

a«

so

as

«to"

a2 ai S^-">

20

- 

30

 r

40 SO 60 70 HEEL AN0LE(d*g)

BUM

COO  (Ion) H01D (lont F.O.I. ( Ion) F.W.I.  lion)

Fig.12

••FORE OS OPEN •»F0REGSCJŒE

» 18 90 79 171 M

t

30

20 SSW

USr-aws. OA FORE GS OPEN •  AFORE GSCtOSE S»N0J CAPSIZED

§600

S S 00 '

I  *00-

calcula tion of Stabi lit y in Flooded Condition for Actual Ship

LU

j |  300.

This figure suggests that if the water in the factory does not f a l l i nt o the fi sh hold, then there is a possibility of prevention of the capsize provided that the hee lin g le ve r i s l e s s than the maximum GZ of about 0. 3 m, but that once f loodi ng through the garbage shoots begins under the co nd it io n of unclosed hatch, then the capsize occurs inevitably.

GZ (m)

ft 2 0 °' 100-

GM=0.6m GM "0.8m

^M

10  20 30 INITIAL r C a ANGLETOSTARBOARD (degj

Fig. 14

HtS. HATCH Ol  Ol 60* MARKS pin, 2  0 STAR, ftOSE. IV 289 A POUT CLOSE m SUB. OPEN >tooo

 ß^J-  typuAlä

Capsizing Test s in S t i l l Water

ira r 'i .J 

i W  3 W

i

"•  J» if   u_  jit» *« ,

'-TI» * '

0.6 0.5 ^5

0.4 0.3 0.2 0.1 0 -0.1

60 70 .. 9(deg.

-0.2

Fig,13. stability with Openings

5. CAPSIZING EXPERIMENTS .  Allowing the flooding through the garbage chutes and   giving the initial heel angle by means of the weight shift, the capsizing experiment was

Photo.2

Example  of Capsizing 'tests in Still Water

(every 45 seconds shot, initial heel  21°)

- I10

Fig,T5 shows t h e e f f e c t of  waves   waves an d  hatch opening o n t h e e l l a p s e t i m e t o t h e c a p s i z e . I t shows that t h e  capsize occurs faster i n  waves than i n s t i l l w a t e r , an d  t h a t t h e c a p s i z e o c c u r s inevitably for the given load con ditio n with  t h e  t h e record unc lose d hatchi Fig. 16 i s an  example of  th of heel i n «aves.

FOfSGSOPE« KOCKOOSE MVEMi'ZS f«f lft.Il«

SS W 

" F i g . f7" f7"  Example of   IfetërLevel 7. Conclusions

1  ;.«sraiW3£K

R e s u l t s o f   e x p e r i m e n t s a n d  s t a b i l i t y calculations confirmed that i f t f  t h e  garbage shoots had been closed, then t h e capsize had never occured under t h e  presumed weather and   load conditions. I t was was conc lude d th a t alt ho ug h th er e were were som somee  t h e occurrence of uncertainties about t h e causes of  th t h e i n i t i a l h e e l an d i t ' s in cr ea se , whic whichh were du e t o t h e wind supposed t o be   p a r t l y due wind for f orce ce s from from the port side, flowing of   f i s h e s i n t h e  factory caused  by the break of wooden par tit io n pl at es , and consequently du e t o t h e o n e - s i d e d h a u l i n g  n e t toward t h e s t a r b o a r d , i n which which t he re ha d  been  du e t o reportedly more than 30 tons of  pollack,  pollack, o r due  the the steering action, t h e  conclusive cause of  the of  t h e  heel t o some capsize after t h e  occurence of t e x t e n t was undou btedl y water flooding through  the unclosed garbage shoots into th e fa ct or y. According According t o th e experiments, i t was al so suggested t ha t even even i f  t h e flooding flooding i nt o t h e f a c t o r y t h r o u g h t h e garbage shoots began, there was a possibility of prevention of  th  t h e  capsize provided that th e  flooded water was confined i n t h e  factory. This should be taken into consideration  in an  emergency like  the present case.

GDI MO'

«x SOD

KUOI  KUOI  a06E INMWE u~INSTUW»ER

(OC GH>0Bm fOS   GSOFE»  GSOFE» \mt  i JKI»2S

. , , * •

i "0

• : CWSXD WS « WS  «FI FIER ER C HU « CF MME »CO« TOCM WV«*2 w in s

10 20 IMTUL HEEL HEEL «NGi£TOSURSOWOOrg)

30

Fig.15 Fig.1 5 Capsizing Tests (effect (effect of wave wav e arid hatch ha tch))

HEEL

1 WO «337 GS:0P£N HAroccLose Ve ts VX"/M INITIAI INI TIAI HEEL 20 Otg. 20 Otg.

SVV\\ SS\^V> A___S_:

SQXn_L— •< m i  rrrrrt-v^—;v\-r-\-_ir—^v-: sas

30

>«

__ __ __

" _3__ÖïS3_Ckl_J_,i.

\jWhJ~* 

 th e c a m i i t t e e , t h e After t h e investigation of  th Ministry of   Transport presented a recommendation on the opening like t h e garbage garbage shoot th at i t  should be closed i n p r i n c i p l e -, especially i n  rough seas and a t th e time of  hauling  hauling  net, and that i t  should be equipped with t h e  remote controllable closing unit o r moved up  above t h e  upper deck.

Fig. 16  Example of   Capsizing Tests in  Wave 6. OTHEREXPERIMENTS Besides above mentioned tests, t h e he el mome moment nt by t h e rudder rudder ac tio n, and by th e  wind force  was  t h e  model drifting i n  the measured. The attitude of  th waves and wind was al so measured. measured. I t was made ade cl ea r t h a t a l t h o u g h t h e  a t t i t u d e o f  t h e  model  was unstable i n beam beam waves on ly , i t  became stable i n wind o r i n   c o e x i s t e n c e o f   wind and  waves and drifted with stable drifting angle from 0 t o 20 degrees depending  on the  heel angle. The relation between t h e q u a n t i t y of   flooded water i n t h e factory and th e  water level i n  various heel conditions was al so observed t o examine  the  t h e  water into t h e  fish p o s s i b i l i t y of   f a l l i n g of  th hold  or the engine rocm rocm ( Fig .1 7).

ACKNOWTJEPGEMEMTS

Authors Authors would would,, li k e t o express th ei r since re g r a t i t u d e t o Profe ssor S.Motora, S.Motora, t h e chairman of the inquiry camiittee of  th  t h e accide nt, Dr.K.Sug ai, the head of  th  t h e subcommittee on the  s t a b i l i t y ,  and of  t h e  committee f o r the ir hel pful all members of t discussion and encouragement. REFERENCES 1. 

\

Morrall,A.,   "The GA  GAUL D is as te r: An  Investigation into t h e Loss of a  Large Stern Trawler",Trans.RINA, Vol.123, 1981 "Report on 2.   Ship Bureau, Ministry of Tran spo rt, "Report Inquiry Committee of  Accident   Accident of   No.28 Akebono Maru", 1982.11

-  11.1 -

Third International Conference on Stability of Snips, and Ocean Vehicles, Gdarisk, Sept.  1986

mmm> Paper 2.5

EXPERIMENTAL RESULTS OF COEFFICIENTS OF ADDED MASSES OF  A SUBMERSIBLE VEHICLE FLOATING UNDER THE WATER SURFACE J. GnieWBzew

• ;-.•-.: . ABSTRACT ,;.'. ••.'• In order to estimate the'coefficients . of added masses of a submersible vehicle with influence of free water surface model. tests have been carried out in the experi mental tank of the Ship Research Institute of the Technical University of Gdansk.In , •this connection linear accelerations . measurements for 13'relative submersions have been performed.The linear acceler ations have been measured for the move ments along all three axes of the.coordi nate system. .  .'.'• 1.INTRODUCTION • ~ '•';,,,. There are several reasons of tactical , and technical nature why submersible ve hicles float close to water surface /es pecially in'the---Baltic Sea .areafwhere the mean depth is 50m/.That is why the problem of the influence of free water surface on .hydrodynamic forces ia of great importance £1] t'[2J ;[3]iThe absence of allowance for the influence of free water surface on hy drodynamic forces in operating submersible vehicles and submarines may lead to loos of motion stability and other related un desirable consequences. The', primary aim of this work has been to determine experimentally the coeffi- . cients of added masses of a submersible •vehicle being in mo tion under water surface at various submersion depths.The aim has been achieved b y measuring linear acceier-. ations for 13 relative submersions.The linear accelerations have been measured for motion along all-three axes of the co ordinate system.Model tests have been con ducted in the experimental tank of the Ship Research Institute of the Technical University óf Gdansk* As the results of.the tests have shown free water surface leads to changes in added masses in comparison with the values of added masses of a submersible vehicle.' floating in unbounded liquid.In this case the quantities of added masses are depend ent on the distance between a submersible vehicle and free water surface as well as , on the direction of a submersible vehicle's motion with respect to free water.surface. 2.RESEARCH PROCEDURES :".'•/'• -•-'. The experimental determination of.the coefficients of added masses óf a hull of a.submersible vehicle was based on,thé fundamental dependence

hence the'coefficient of added mass

Kli

m

Por the purpose of accurate estimation of the values of added masses of a vehicle the force making the vehicle move was'being changed.A coordinate system bound with the. center of the vehicle's mass  /fig.1/ was assumed her e. -.

Fig*1 Coordinate axes system óf a sub mersible vehiole i n determining added masses. The tests consisted in measuring acceler ation values of a submersible vehicle. The model tests were conducted in the experimental tank of the Ship Research •. Institute of the Technical University of Gdansk.In order to conduct the tests a spécial measuring devioe had been designed to measure accurately linear accelerations in the case of linear, movements of the model vehicle.The following scheme  /fig.2/ was applied to determine linear acceler ations. :;.

•;-::';;-\;?t^'^';i.-^;'- ..-'"ƒ• hi ' where

F^ - external force acting on à . : submersible vehicle.; '•; mi. - total of vehicle's mass and of yehicle's added  mass, ,Ai'i ' a,. - acceleration of vessel's  mass. Thus equation  /.1/ 'may be written as m +A li

Pig.2 Scheme of the measuring device for determining linear acceler :; ations.

I .

-.113

The'.mean values of the coefficients -of added masses obtained from lha nóiSel tests are presented in tables /4,5,6/ and in diagrams /fig.3,4,5/.

A simplified gravitation device i s presentin the scheme.Accelerated linear motion of the model i3 made by the falling of a weight of an appropriate mass caused by gravitation.In this case the resistance of the device was
40B  aem

 Kn

 oou

 a*4 AMI OjMO 4MT

L est 0.760m B St 0.063m . H S3 0.063m 3 1.572kg M a J o - 0.04571 kg m' scale 7t S 100

SE

4M«

 apt*  Hot

The model was made of wood and covered with polyurethane paint.Its buoyancy was neutral.

an

uu

««

V»

4M

Fig.3 Values of the coefficients of added mass of the submersible vehicle floating along OX axis at different submersion depths obtained from the model tests..

Symbols used in a tD./3 2 J B r.nii F CHI. H GK>1 L [m] K ii

the paper acceleration width of the model towing force height of the model length of the model coefficient of added mass . M a (kg) ; 7 mass •. • , V |Wö] -velocity of the model Ali Heg] - added mass scale of the model 1 = Y V - relative submersion Y - submersion of the Lm] model 3 ; ANALYSIS  OP-THB RESULTS Linear accelerations were measured. for 13 relative submersions of the model vehicle.They were measured for motion along all three axes of the coor. dinate system.Since the direct effect of each of the measuremente mentioned above was measurement of the distance travelled by the model of the vehicle in function of time the first stage i n computing pro cedure was determination of the velocity of the model in function of time,that is V„ a f  ('t).It was that curve determined off the ba sis of regres sive analysis whic h became the basis for determining the ac- • celeration at the initial point.Since the regressive analysis made with the KP-9830A computer ha« shown that the curve is de scribed by quadratic polynominal,that is

\K*t (00 Oft

4M 090 OjK 004

m 0.00.

0,10

0/0

OßO

 au

Pig.4 Value s of the coefficients of ' added mass of the submersible vehicle floating along OY axis at different 'submersion depths obtained from the model tests.

V/t/ =. at 2 + bt + c

thus differentiating V/t/ with 'respect to time at point 0 ' in i ti a l acce lera tion dV

 ojo

= 2 at + b = b

1B  obtained.Thus coefficient b determines. the acceleration value at the initial. point. Regressive analysis of each velocity curve /in other words.of each load/ was made with the HP-9830À computer. The reults are'presented i n tabular form as values of coefficients of added mass in function of relative submersion, that is K,, => f/t/ for each of the loads /tables 1,2,3/.

Pig.5 Values of the coefficients of. . added mass of the submersible vehicle floating along OZ axis a£ different submersion depths obtained from the model tests.

114

on

TABLE 1 F

i

0.19 6 a

' • t

0.10 0.11 0.12 0/13 0.14 0.15 0.16

; ©.ie 0*20 '''-'.  6.23 0.30 v 0,40 0.50

i

' 11

0.1233 1v5Ö95 0.1233 1.5899 0.1232 1.5904 0.1232 1*5908 0.1232 '1.5912 0.1232 1.5916 0.1231 1.5920 Ó.1231 1.5927 0.1230 1.5934 0.1229 1.5948 0.1229 1.5958 0.1226 ,1.5966 0,1228 1.5968

>11 0.0175 0.0179 6.0184 0.0186 0.0192 O.ÓI96 0.0200 0.0207 0.0214 0.0228 Ó.0238 0.0246 0.0248

0.4 90 fi

2 0.3086 0.3085 0.3084 0.3083 0.3083 0.3082 0.3081 0.3080 0.3079 0.3076 0.3074 0.3072 0.3073

m* Aj 1

0.686 ^11

1.5877 0.0157 1.5882 1*5887 1.5891 1.5895 1.5899 1.5903 1.5910 '1.5917' 1.5931 1.5941 1.5949 1.5946

0.0162 0.0167 0.017I 0.0175 0.0179 0;0183 0.0190 0.0197 Ó.0211 Ó.0221 0.Ó229 0.0226

.':/ •'*5- :•••

ma it

A 1r

0.4323 Ó.4322 0.4321 0.4320 0.4318 O.43I7 0.4316 0.4331 0.4312 0.4309 6.4306 0.43Ö4 0.4305

1.5868 1.5873 1.5877 1.5881 1.5886 1.5890 1.5893 1.5901 1.5907 1.5921 1.5931 1.5939 1.5934

0.0146 0.Ó153 0.0157 0.0161 0.0166 0.0170 0.0173 0.0181 0.0187 0.0201 0.0211 0.0219 0.0214

TABLE  2;.•;,

• 

••>i--*'-

:

;.'. N'.-..-:;

'?'.;fv '0.10 0.11 0.12 0.13 0.14 0.15' 0.16 0.18 0.20 0.25 0.30 0.40 O.SO

•:'•••';•-.;

''•••.*»•?:•:

0.1696 0.1691 0.1687 0.1682 0.1678 0.1674 0.1670 0.1Ô63 0.1652 0.1645 0.1636 0.1632 0.1644 -i I

0.490

•.'•/';

0.981

m* A 22

*22

2.8893 2.8974 2.9052 2.9127 2.9199 2.9268 2.9334 2.9437 2.9668 2.9794 2.9946 3.0025 2.9807.

1.3173 1.3254 1.3332 1.3407 1.3479 1.3548 1.3614 1.3737 1.3948 1.4074 1.4226 1.4305 1.4087

- ' a 2 . ' 

•''

0.3416 0.3406 0.3397 0.3388 0.3380 0.3372 0.3364 0.3350 0.3337 0.3312 0.3295 0.3266 0.3311

1.471

m + ft 22  /^22.;.'.' V •? •;...'• m+ A»? 2.8719 2.8800 2.8878 2.8953 2.9024 '2.9093 2.9160 2.9283 2.9394 2.9620 2.9771 2.9851 2.9632

v;1Ï5v--

1.2999

1.3080. 1.3158 1.3233 1.3304 1.3373 1.3440 1.3563 1.3674 1.3900 1.4031 1.4131 1.3912

O.5154 0.5139 0.5125 0.5112 0.5099 0.5087 0.5076 O.5054 0.5035 0.4996 0.4971 0.4957 0.4977

2.8541 2.8622 2.8700 2.8775 2.8847 2.8916 2.8982 2.9105 2.9216 2.9442 2.9593 2.9673 2.9555

A

22

1.2821 1.2902 1.2980 1.3055 1.3127 1.3156 1.3262 1.3385 1.3496 1.3722 1.3873 1.3953 1.3835

TABLE 5 0.490

' • ' .

. * ' " '

• • . . '

>1

m\ 3

0.981

À53

B

"2

**33

I.47I

"**33

*33 •

o. id 0.11 0*12 0,13 0.14 0.15 0.16 • 0.18 0.20 0. 25 0.30 0.40 0.50

0.1507 0,1504 0.1501 0.1499 0.1496 0.1493 0,1491 0.1487 0.1485 0.1475 0.1470 0.1467 0.1467

3.2313 3.2578 3.2640 3.2700 3.2758 3.2812 3.2865 3.2963 3.3050 3.3227 3.3343 3.3392 3.3397

1.6793 1*6858 1,6920 1.6980 1.7038 1.7092 1.7145 1.7243 1.7330 1.7507 1.7623 1.7672 1.7677

0.3156 0.3149 0.3143 0.3137 0.3131 0.3126 0.3120 0,3111 0.3102 0.3085 0.3074 0.3069 0.3059

3.1086 3.1151 3.1213 3.1273 3.1330 3.1385 3.1438 3.1535 3.1623 3.1800

,3.1916 3*1965 3.2070

1.5366 1.5431 1.5493 1.5553 1.5610 1.5665 1.5718 1.5815 1.5903 1.6Ö80 1.6196 1.6245 1.6350

•

*

3

0.4683 0.4674 0.4664 0.4656 0.4647 0.4639 0.4631 0.4617 0.4605 0.4579 0.4563

,0.4556 0.4569

3.1410 I.569O 3.1475 1.5755 3.1537 1.5817 3.1597 1.5877 3.1654 1*5934 3.1709 1.5989 3.1762 1.6042 3.1859 1.6139 3.1947 1.6227 3.2124 1.6404 3.2240 1*6520 3.2289 1.6569 3.2194 1.6474

TABLE 4

[N) f

-

0.10 0.11 0.12 0,13 0.14 0.15 0.16 0.18 0.20 0.25 0.30 0.40 0.50

Ni

K

11

-Ni

K

0.0111 0.0157 0.0179. 0.0114 0.0162 0.0184 0.0117 0.0167 0.0188 0.0120 0.0171 0.0192 0.0122 0*0175 0.0196 .0,0123 0.0179 0.0200 0.0|27 0.0183 0.0207 Oï0132 0.0190 0,0214 0.0136 0.0197 0.0228 0.0145 0.0211 0.0238 0.0151 0.0221 0.0246 0.0156 0.0229 • 0.0248 0.0158 0.0226 0.0175

.'' •  "".'

.  0.686

0.490

0.196

*33

•

11  ;

0.0100 0.0103 0.0106 0.0109

OéOin 0.0114 0.0116 0.0121 0.Ö125 0.0134 0.0141 0.0146 0.0144

: -  116 ^

*114

'•h-i _

^1 sredn.

0.0094 0.0102 0.0148 -i .. 0.0153 0.0097 0.0105 0.0100 0.0108, 0.0157 0.0161 0.0103 0.0111 0.0166 0.0105 0*0113 0.0170 /Ó.0108 0.0116 0.0110 0.0118 O.OI73 0.0181 0.0115 0.0123 0.0187 0.0119 0.0127 0.0201 0.0128 0.0136 0.0211 0.0134 0.0142 0.0219 0.0139 0.0147 0.0136 0.0146 0.0214

TABLE 5 0.490

cm .•'.'..it  '-:

Hz

<*22

0.40 1.3173 0.11 1.3254 0.12 1.3332 0,13 1.3407 0.14 1.347? 0.15 1.3548 0.16 1.3614 1.3737 Oiia 1.3948 :o.2Ó 0.25 ; 1.4074 1.4226 0.30 1.4305 0.40 1.4087 0.50

0.981

0.83Ö0 0.8431 0.8481 0.8529 0.8574 0.6618 0.8660 0.8739 0.8873 0.8953 0.9050 0.9100 0.8961

*22

1.471

^22

A

k

22

•

1.2999 0.8269 1.2821 1.3080 0.8321 1,2902. 0.8370 1.2980 1.3158 0.6418 1.3055 1.3233 0^8463 1.3127 1.3304 Ö.8507 1.3196 1.3373 1.3440 .0.6550 1.3262 0^8628 '1.3385 1.3563 1.3674' 0.8698 1.3496 0.8842 1.3722 1.3900 0.8938 1.3873 1.4051 0.8989 1.3953 1.4131 0.8850 1.3835 1.3912

•

•

*

&

;

.

0.490

f 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.18 0.20 0.25 0.30 0.40 0.50

«33

1.6793 1.6858 1.6920 1.6980 1.7038 1.7092 1.7145 1.7243 1.7330 1.7507 1.7623 .1.7672 1.7677

1,0683 1.0724 1.0763 1.0802 1.0838 1.0873 1.0906 1.0969 1.1024 1.1137 1.1211 1.1242 1.1245

*33 1.5366 1.5431 1.5493 1.5553 1.5610 1.5665 1.5718 1.5815 1.5903 1.6080 1.6196 1.6245 1.6350

:

 

22 sredn. , 0.8268 0.8320 0.8369 0.8417 0.8463 0.8506 0.8549 0.8627 O.8719 0.8841 0.8938 0.8988 0.8871

TABLE 6

0.981

^33

•

0.8156 0.8207 0.8257 0.8305 0.8351 6.8394 0.8436 0.8515 0.8585 0.8729 0.8825 0.8876 0.88Q1

I

CN]

•

1.471 K

33

*33

0.9775 1.5690 0.9816 1.5755 0.9856 1.5817 0.9894 1.5877 0.9930 1.5934 0.9965 1.5989 0.9999 1.6042 1.0060 1.6139 1.0116 1.6227 1.0229 1.6404 1.0303 1.6520 1.0334 1*6569 1.040,1 ,1.6474

•f-.;ii7;--

K

33

0.9981 1.0022 1,0062 1.0100 1.0136 1.0171 1.0205 1.0267 1.0323 1.0435 1.0509 1.0540 1.0480

K

33

éredi».

1.0146 1.0187 1.0227 1.0265 1.0301 1.033,6 1.0370 1.0432 1.0488 1.0600 1.0674 1.0705 1.0709

4,CONCLUSIONS A few general conclusions based on . the -tablés and diagrams obtained from the model tests have been 'formulated below: L Fr e e water surface leads to changes in added masses in comparison with the values of added masses of a submersible vehicle floating in unbounded liquid. 2.The quantities of added masses are dependent on the distance between a submers ible vehicle and'free water surface /on the submersion depth/,as well as on:the direo*' tion of a submersible vehicle's motion with respect to free water surface,and is not dependent on the velocity of a vehicle. 3.The values oî  all coefficients of added masses of à submersible vehicle de crease with the decrease of the relative submersion depth  7,thus free water surface leads to a decrease of the values Of added masses of a submersible vehicle; .4.The influence of free water surface on added masses of a submersible vehiole decreases with thé increase of the relative submersion d e p t h T and is practically nonexistent at the/relative submersion depth  f>0.5.  ••' 5.A considerable influence of free ;v/ater surface on added masses occurs at the relative submersion depth ? =>  0.10.This in fluence causes the values of coefficients of added masses of a Bubraersible vehicle to decrease approximately by 50 #• of the values of coefficients'of a vehiole floating at a large submersion depth /i n unbounded liquid/.

- 118 -^

RBffBRflNCES 1.GNIEWSZEW J.tWpiyw swobodnej powierzchni wody na aile, noéna. okrçtu pod-, wodnegö przy plywaniu w poiozeniu podwodnym.Zblór Prac VJSMW.Gdynia '. 1972;no.36,105-124. 2.GNIEWSZEW J.:Opór okre/tu podwodnego poruszaja_cego sic w 'stanie' zànurzonym w pobliéu swobodnej powierzchni wody. Studia i màterialy oceanologiczne no.20.0ceanotechnika/2/,ÏAN,So'pot 1977. 3.GNIEWSZEW Jï,:Wpîyw swobodnej powierz chni wody na wzdiuzny moment, hydrodynamiczny-.okre.tu podwodnego przy p iy waniu w poiozeniu podwbdnym.Zeszyty Naukowe WSMW,Gdynia 1975;no.1/44, 37-55. 4.JAR0SZ A.:Przegla,d i ocena stanowisk > badawczych stosowanych do badan modelowych ciai zanurzonych.'lnstytut Okretowy PO,Gdansk 1976;no.52.

r

 Mmw>

Third International Conference on Stability of Ships and Ocean Vehicles. Gdarisk. Sept..1986

Paper 3.1

 A CRITERION FOR SHIP CAPSIZE IN BEAME SEAS H. Sadakane

ABSTRACT This paper deals with a ship capsize in beam waves during one-swing from weather side to lee side. A formula for a critical capsize condition is found on the basis of an unsteady-roll analysis using an apploximate solution technique of simplified roll equation. The condition can be applied and extended to the effect of the wave period, the shape of GZ-curve , the roll damping on the ship capsize. etc: 1 .  INTRODUCTION

were made by Froude [1 ], Watanabe [2] and Kato [3]. They clarified the basic charac teristics of linear roll in unsteady  pro cess. Subsequently, Krappinger [4] calcu lated a capsize condition by taking into account the variation of ship restoring moment in following waves. Recently Blocki [5] has made interesting works with respect to the ship capsize due to thé variation of ship restoring moment. . In this paper, a theoretical analysis is made on a critical capsizing roll during one swing from weather side to lee side, using a simple roll equation with nonlinear

It has been recognized that ship cap size is caused by many factors and by their combinat ion, like wind , shipping of water on deck, breaking of cargo, breaking waves, an abrupt change of current, coupling of snip motion, and so on. However, the ca p size itself does not seem to have been clar ified enough in view of its dynamics, since the capsize can be considered as an unsteady dynamical process of the extreme roll.  It would be therefore necessary make consider ation the extreme roll process by using analitycal and/or numerical treatment. If the capsizing phenomenon, represent ed by a single equation of roll motion, is defined as a situation that, during a pro cess of increasing roll amp litude, the roll , angle exceeds the vanishing point of stabil ity, then the capsize condition and associ ating factors may be found by analysing the process in detai l. Such a deterministic process probably affected by three factors, namely the ship properties like'stability, Inertia and damping, the ; wave properties like period and steepness, and finally the initial condition of the roll motion. In early tim e, theoretical analyses on the concept of increasing roll amplitude

restoring term and an approximate solution method. As a result, a formula for a clitical capsize condition is found, which is expressed in terms of the three factors men tioned .above. Secondly as an application of the critical line, discussions are made on the critical value of the wave period as a most probable period for capsize and the effectiveness of the ship stability factors against the capsize. 2.

EQUATION OF ROLL MOTION AND SOLUTION PROCEDURE

The following equation of apparent roll motion with nonlinear restoring moment is used to study the present ship-capsize problem. i

i

ë +2K„ê-+».f(ej,)»(Y*H/A)&> sin tot a e a « where

f(eJ = e Ä +B se +$56+676^ ---

(1) •-

(2)

In the above,- "a represents the apparent roll angle, K the linearized damping coefficient, w 0 the natural frequency, y the coefficient of effective wave slope, H the wave height, A the wave length, u the wave frequency and t the time. The superscript dot " '"stands for the différenciation

- .119 -

with respect to t, and P„ (n>3,5,•--) the coefficients for the nonlinear restoring ncraent. Supose the roll amplitude and the roll* period vary with time in an unsteady' capsizing process. Then a phase plane analysis nay be appropriate for examining the amplitude change ( 6) . For variables on  phase plane, we adopt here the roll ampli tude A and the phase difference $ between the exciting moment and the roll angle. In order to compose the phase plane expressed  by the terms A and    should be transformed into a set of simultaneous ordi nary differential equations of first order. 2.1

Averaged equation of roll motion

(4)

0

VÔa

.I + (-yiTH/X)w:is1n wt

J

(5)

From eqs. (3) and (4), the following equations can be obtained.

 A 2 =e a +(e a /w)

(6)

TanUt-^Me./eJo) a a Differentiating eq.(6) with respect to t and rewriting it using eq.(5), we can obtain the following equations. +(fnH/X)u sin ut]cos(ut-+)

(7)

••(l/ultu'sIntut-^J+^-ÎKg^-u.ftej) + (YitH/X)u>2s1n ut)]s1n(ut-*)

Taking the average of eq.(7) over the peri od 2ir/u , and considering the roll amplitude  A and the phase difference 4> to be constant during the period,performing the averaging, we obtain the so-called averaged equations of roll motion in the following form.  Â=-K A+y(Y7rH/A)«s1n $ e

Where

2

2

Z

2

I

2

2 2

{2K A  e B ) +Ao (-« +a>» g(A»)) =(YITH/X)IÜ  |H)

(12)

where Ts  (=2IT/Ù>S) is the roll period for A, , and To(=2n/(i>o) the roll period for an infinitesimal roll amplitude, i.e., for the linearized motion. 2.3

Denoting é as 9 eq.(1) becomes the a a following set of first-order equations

0 a =-2Ke ê a-uo f(e a )

Substituting JUo and <>=0 into eq.(5) and (6) and eliminating $ , we obtain the steady-roll solution

Ts =T e//g(A  0)

where A and 0 are also assumed to be slowly varing functions of time t, i.e. À « Au) and  A^«An). Then the différenciation of eg. (3) with respect to t can be expressed as

2

In a extreme roll, since the natural roll period of a ship changes with the roll amplitude , the relation between the amplitude and the period plays a important role to the roll of the ship in waves. In this section, the backborn curve describing the above relation will be derived.

(3)

6 #Au> cos(n>t-$)

Backborn cu rve

where A„ is the steady-roll amplitude. Eq.(11) gives the frequency-response curve. Substituting  (YUH/X)IO =°» K e=° and w=
 An approximate solution of eq.(1) can  be derived by assuming 6 =A s1n(u>t-)

2 .2

Unsteady roll and phase plane

For the preparation of the analysis on the ship capsize, the characteristics of the unsteady roll is discussed by the use of the phase plane. Fig.1 shows an example of trajectories of motion (  =0 on the  phase plane, a contour linked through these  points gives an information on the behavior of the trajectories, i.e., the unsteady roll. Such contours' are calculated by the following equation. .  A=(YuH/X)(ui/2Ke)s1n •'

(13)

(8) <).= cos*1{A(ü)og(A)-u) )/(YirH/X)iu } 2n

  g(A) = l4 0 8 A + -2{|B±|Hß 2 n + 1 A   (10) n = l , 2 , 3 ••••

- 120

(14)

Eq.(13) gives the contour of À=0 as shown in Fig.1 The inside domain of this contour corresponds to A>0 . Eq.(14) also gives

the contour of $=0 as shown in Fig.1.  The domain of $>0 is the upper side of this contour. Accodingly if the contours on the phase plane are prepared, we can approxi mately predict, the course of the trajecto ries,  i.e. the subsequent roll motion. 3.

CRITICAL CONDITION FOR CAPSIZE

For the criteria of ship capsize, the lee-side heel angle for judging ship cap size has usually'been calculated using the energy conservation concept in unresisted roll in still water. However,the roll damp ing and the exciting moment do act on a ship hull while it swings the weather side to lee side. These moments will rationally be tak en into account in the present analysis by considering the nonlinearity of the restor ing moment and the unsteady roll in waves.

Considering the one swing capsize from weather side to lee side, the capsize process should be initiated by the weather-side roll angle 6 and the phase * aw * difference Q . Therefore, if we plot the points ( ,6 ) in extreme rolls, we can get a diagram dividing these points into two region, i.e., a set of points which lead to capsize, and the other uncapsized, as shown in Fig.2. The hatched domain in Fig.2 corresponds to the group of capsized points, in which the lowest point 6 ,„ should represent a critical situation for capsize to occur at a least possibility. In other w ords, if the roll angle of a ship

In this chapter, as the first step of considering the above problems, the capsize due to waves will be analyzed by using the preceding approximate solution technique, and a critical condition for capsize will be stated. Capsize ' is defined here as the condition that the roll angle of ship exceeds to the vanishing point of stability and the waves are assumed to be sinusoidal during one swing of the ship roll. 3.1

Factors of capsize To describe an equation of capsizing roll motion and its critical condition, it is necessary to find out basic factors in the unsteady roll process using a simplified roll equation. The factors will be found in the condition of increasing roll amplitude and in the initial condi tion of weather side roll,  as follows. (1) Condition of increasing roll amplitude In order for the roll amplitude to increase during one swing, it is necessary that (yirH/X)ü)/2 representing the exciting moment is large and $ becomes about ir/2 during one swing, which can be understood from eq.(8). At the same time, the condi tion. $>0 also makes the roll amplitude increase.  Since the condition $>0 makes the phase velocity oi-i of the roll angle 8 a decrease* the roll period 2ir/(u>-4>) becomes large. The prolonged time brings the increase of the roll amplitude because, the condition Â>0 during one swing. , (2) Initial condition of weather -side roll angle - 121 -

in a wave ex ce ed s th e e

, de pe nd in g ön w

, there is a possibility that the ship may capsize after one swing to the lee side. Accordingly, the fl can be considered as awm a factor, for the capsize. And the phase f > can be taken as 2/it approximately. angle ( From the above discussion, two kinds of factors for the capsize can be proposed, namely one is (yïïH/X)u) , the other is9 a w m • 3.2

Determination of capsize condition

Studies on the capsize in waves have so far been carried out through experiments and numerical simulations. Considering the complicated mechanism of the capsizing phe nomena, these direct methods are still necessary. To understand the phenomena physically, however, theoretical approach should be needed. In order to derive an expression for capsize cond ition, the capsize is assumed he re, as us ual, that the roll angle exceeds the vanishing point of the ship stabili ty6v . . The condition for capsize is expressed as the following inequality.

l 8 a w J + A A > V 

(15)

where |8 m | represents the amplitude for awm' a n<^ A'A t n e increment of A during one swing from weather side to lee side, which can be approximated to be the form.

8

AA*ÀT

H«)

As shown in Fig..3, the time derivative A is calculated at the time whe n A=( 16|+8v)/2 and T is the one swing period. T is estimated: from the phase velocity mentioned in the section  3.1(1). T*Tr/(w-$)

<17)

When the roll period in nonlinear and in unsteady roll is concerned,
ed À« Au> arid At «A») to obtain the first ap  proximate solution in the section 2.1. Eq. (17) is approximated to the follow form.

u(yz)ti4/»>

e

where T w the wave period. For  and $ in eqs. (16) and (18), the following equation can he used which is derived by substituting $=w/2 into eqs.(8) and (9).

.

2

v  A=(le » awm'I + 8  v ")/2

4.

(21)

4.1

> 2K A/( e w ,e ¥ )4i£{3-g(A)/vV (i-|e awm |/e v) 

(22)

where V denotes the tuning factor. The right hand side of the expression (22) includes the roll damping, the stability of the ship, and the initial roll angle in the weather side. If the exciting moment in the left hand side exceeds the terms of restoring moment, damping moment etc. in the right hand side, there must be a  possibility for the ship to capsize.

J

The nonlinear roll damping can also be considered for an extreme roll. When the roll damping term in eq.(1) is expressed as

2(K,éa+K2|éa|êa+Ksèa3) the equivalent damping coefficient K in the 2nd term of the right hand side in eq.(22) should be replaced to the following form. K. = K1+-|trK A+|K 2 ])u)o A

3.3

(23)

Critical line for capsize

From eq.(22) we can define a critical capsize line which separates a non-capsize domain and the domain of possible capsize, in the following form. I ^ v

APPLICATION OF CRITICAL CAPSIZE CONDITION

The preceding discussion on the criti cal capsize condition can also be extended to other aspects of ship stability.

Rearranging eq.(15) using eqs.(16) through (20), and dividing by w„8 , the condition for least possible capsize becomes the form, I ^ v

(26)

capsize happens immediately, since the roll angle can not recover again to the upright  position because ®  B £0. Such a value of the point P may play a simple index to es timate the ship stability against capsize.

(20)

•-(w -we g(A)}/2a> where

JfZHAv > Ke/*„*i£<3-g(A)/vY

(19)

2

U5>

one can see that no capsize happens. Conversely, for a wave height exceeding a  point P as shown in  Fig.4, naaely

«18)

 A»-K A*£(ywHM)u> e

< 2K /wo

- 2K A/(w e 0 9 v )+i£{3-g(A)/vY

0-|e a w J/e v ) (24) For instance, as shown in Fig.4 , the critical line can be drawn by taking the ordinate and abscissa as l 9 a w J/ e v a n d   (Y*HA  A)V/8 respectively. When the wave height is small enough to satisfy the condition, - 122

Tuning factor and capsize

The peak amplitude in steady roll  motion of a ship takes place at the reso nance wave period on the frequency-responce diagram. However, the resonance wave per iod in unsteady process like capsizing can not be defined clearly, since the roll  period of changes with the roll amplitude as seen in the preceding chapter. However, we can find a particular value of the wave  period concerning capsize. The right hand side of eq.(22) can be regarded as a func tion of the tuning factor V , and becomes a  minimum at a certain value of V,  i.e.V  c. The value corresponds to a critical period which induces a capsize at the smallest wave steepness. From eq.(22) the value of such a critical tuning factor can be obtained in the form. vc=/g(A) where

(27)

A=(|6 awm'I+6 v)/2 "

Considering the right hand side of eq.(27) is the same as eq.(12) giving the  backborn curve, the wave period  v cTo corresponds to the ship roll period with the mean roll amplitude (l6 I+6 )/2. Thus • aw m' v , eg.(27) gives a critical wave period for a ship to have the most possible occurrence of capsize. By comparing the critical capsize line for the v_ value, we may 'discuss the effect of, for example, shape of GZ-curve on capsize, which will be stated in the next section (2). 4.2 Critical capsize condition and ship stability The present critical condition is

affected by the ship stability. (1) G Z m a x and the stability GZ r i.e., the maximum value of GZ,  max •is an important factor related to the ship stability. It is said that higher GZ w ID3 X

 brings better stability. Let us examine this from the point of the critical capsize three kinds of GZ-curves GZ. ,, GZ and condition calculated by eq.(24).  A-2 Suppose  A-l ' GZft.3respectively, as shown in tht left in values is 3:2:1 . Fig;5. The ratio of GZ  max The corresponding critical lines are clearly influenced by the difference of G Z m a x . We can interprète this difference as the likelyhoöd of capsizing in waves. For example, taking the wave of (YHH/X)V/6 1/6 in =0.8, we can read the value of In 1 1  awm  ' v Fig.5 as follows. 'eawml'ev " °-4 I ,28) For For GZ-curve GZ-curve A-2 A-3 •> ,. K  = 0 I The.ship having GZ-curve A-3 will capsize even from upright, while the ship having GZ-curve A-2 is not the possibility of cap size within about 40 % inclination of the stability range to the weather side. The signs • , % and A in Fig.5 represent the points calculated by a numerical simulation using Runge-Kutta method. The fact that the agreement between them is quite well indicates a validity of the  present critical line. (2) Shape of GZ-curve and critical capsize condition The effect of Bhape of GZ-curve on capsize has so far been examined only in . view of dynamical stability. However, this effect can also be studied by using the  present critical line method. In order to compare the critical lines corresponding to three kinds of GZ-curves as shown in the left in Fig.6, in which the the values of dynamical stability at 8 are all the same, we can find that a round GZ. curve like C-1 has poor stability against capsize, as shown in Fig.6. In the figure, the tuning factor is prescribed to  be the severest value, l.e.,v in eq.(27). 5.

EXPERIMENT

To confirm the results obtained in the  preceding chapters, capsize experiment is carried out in a water tank. Two kinds of cylindrical models are used of which the cross sections are ellipse and square, called elliptic model and box model respectively, as shown in the right part of Fig.7. GZ-curves for these models are

considerably different in form and GZ ..so  max' that the stability of the box model is very  poor. Experimental conditions are listed in Table 1. In the experiment, regular beam waves are generated. The model is set in the water tank making its lateral motions free: To establish the initial condition $w and 0 the model's roll is restricted at the begin ning to keep heel in certain angle, and then at the prescrived wave phase angle,the  model is made free from the restriction. Experimental results are shown in Fig.8. The marks • and % are the experi  mental points of the box model and of the elliptic model respectively. It is shown that the experimental points for the box  model , having even poorer stability,lie in safty side for capsize rather than those for the elliptic model. Calculated results  by eq.(24) agrees qualitatively well with the measured data. 6. CONCLUSIONS  A critical condition for ship capsize in beam waves is proposed on the basis of the unsteady-roll analysis using an approx imate solution techniqe of simplified roll equation. The critical capsize condition can be expressed in terms of GZ-curve, roll damping coefficient, tuning factor, wave steepness, and initial roll angle in weath er side.' The present critical condition will also provide an useful criterion for the ship stability against capsize.  ACKNOWLEDGEMENT The author wishes to express his deep appreciation to Professor N. Tanaka and Professor Y. Himeno, at University of Osaka  prefecture, for their instructive discus sions and guidance through the present work. The author also wish to thank to  Mr. K. Yamamoto for his active assistance in carrying out this investigation.  NOMENCLATURE  A Roll amplitude  A Steady-roll amplitude GZ Righting arm H Wave hight T One swing period T Roll period in small amplitude T . Roll period in finite amplitude t Time 8 Coefficients for nonlinear righting arm Y Coefficient of effective wave slope 'a

e

aw

Initial roll angle in weather side

"aw» Critical roll angle of

6a

6y Vanishing point of stability K e Equivalent damping coefficient K1..Nonlinear damping coefficients (l=1,2,3) X Have length v Tuning factor v c Critical tuning factor (J> Phas Ph ase e diff di ffer eren ence ce *w o w s u

Initial phase difference in  in weather side Natural frequency in small amplitude Frequency in finit amplitude

REFERENCE

11] Wi Froude:The Non-uniform Rolling Ship, Trans.R.I.N.A., Trans.R.I.N.A.,  1896.

of

(2) Y.  Watanabe:On the Limiting Angle of  the Limiting Roll on Irregu  Irregular lar Waves , Jour.S.N.A.J., Jour.S.N.A.J., No.50,   1932. [3] H. Kato:ON the  Ships among  the Rolling of  of Ships Irregula Irregular r Wave s, Jour.S.N.A.J., Jour.S.N.A.J., No.65, 1940. [4] O. Krappinger:Uber Krappinger:Uber Kenterkriterien , Schiffstechnik 9-H. 48,  1962. 15] W. BlockitShip Safty in  in Connection with Parametric Resonance of the Roll,I.S.P. ,No.306,  1980. [6]  A. Y. Odabasi:A Morphology of  of Mathemat ical Stability Theory and its  Applica tion to Intact Ship Stability Asses s ment, Second ment, Second Int. Conf. on Stability of Ships and and Ocean V ehicles, Tok yo,  1982.

Table 1  Experimental condition

To

Model Elliptic

model

0.785

Box model

2.50

».

K. (1/MC) 0.43

0.7-0.9

1/10-1/15

0.32

2.0,2.5

1/18,1/10

GZ curve Eq.0 «) Col.

K.IW..0.ra  V  • ' • '

e,

HIK

•

•

t . | A-l

 A

A-J

1.0

Roll amplitud«

Xoflt  

Ä

V 08  1.0 fHH/Xu

 on critical Fig. 5. Effect of GZmax on capsize condition \

iX

3««

,4

0.4  0.6

0.2

Bacfcbom curves

Phase dfterence • . Fig. 1 .  Example o f  phase p lan e diagram J  B-l

J**

r J-*  M.À.  M.À.

Capsize Roll amplltudt

«

Fig.

2.

 Xy

No capsize ni.i 

Effect

o f 

fc.A

en na mi Initial phase dMtrence • .

~;

ta 

i n i t i a l c o n d it i t i o n f o r capsize

JE2HÜLO V 6v

T.;t,

Fig. 6. Effect of shape of GZ-curve on critical capsize condition

Capsize

Elliptic mod t l

0

• - t M —H utHcm

Fig.

Fig. 3. Roll angle during one  one swing

  Model sections

7

20 40 Heel Heel «igle (tftg)

e«p.

lOcr

'»y  

1.0 1.0 T

Possblllty of  capsize

th ei r GZ-curves GZ-curves

and

Elliptic model 001

\.

model

Eq.(M)

• • ~.=

 r 0.60 0.86

^:

0.5No capsize

-t a<

^

•" '

Fig.

4.

0

8,

0.2

(Wove (Wove height hei ght))

0.4

0.6 0.6

0.8

1.0 1.0

12 12

e.

Fig. 8 Critical capsize condition for cylindrical cylindrical model

Crit ica l capsize lin e -  124

Third International Conference on Stability of Ships and Ocean Vehicles, Gdansk, Sept. 1986

OJV

O )  (OlVo

Paper 3.2

STABILITY CRITERIA FOR PRESENT DAY SHIPS DESIGNS H. Hormann, Hormann, D. Wagner Wagner

ABSTRACT

2. CHANGE IN IN THE RANK. RAN K. WHICH STABILITY STABI LITY HAS

During the past twenty years a dramatic change has taken place in the design of dry cargo ships, which has a bearing upon their seaway behaviour and thereby upon their stability characteristics. The traditional methods for assessment of the stability of ships take these changes into account to a limit ed extent only and may entail misjudgements of sta bility, as has been proved by tank tests. The intro duction of a form factor "c" has made it possible to assess stability in such a way as to take into account individually and in line with their signifi cance the parameters determined by tank tests, such as principal dimensions, KG and ship's form. The application öf this new method of stability assess ment has a bearing upon the minimum stability   val ues required. For instance, with an increasing B/D ratio these increase as against the requirements of IMO Res. A 167. However, adherence to these more stringent "weighted criteria" only leads to a safe ty level equivalent to that of "traditional" ships, but not to a higher one. At the example of three ships with widely differing B/D ratios the effect of this new method upon the stability requirements is demonstrated and compared with the requirements having existed so far.

Parallel to this brisk change in the transporta tion system, the determination and assessment of in tact stability have become continuously more impor tant. In former days, particularly with big ships, stability was a natural side product, so to speak, generally not requiring any particular attention in the design phase. Nowadays, however, the proper de termination of stability has become one of the most important design parameters on which a vessel's eco nomic success or failure essentially depend. It is evident that previously existing stabili ty reserves are nowadays being utilized and that ships are loaded up to their stability limits. Now, where is this limit ?

1.   INTRODUCTION

In the early seventies, the container was introducted into shipping as a means of transport. At that time, sceptics predicted not much life expec tancy for this trend. Compared with conventional ge neral cargo traffic, t n e investments and extra orga nizational efforts required appeared to bé extraor dinary so that it was deemed doubtful that this no vel transport system would prove to be an economic success. Today, we know that, in fact, the develop ment was completely different. The container has re volutionized shipbuilding and shipping and led to ships being constructed that are capable of carry ing deck loads which would have been inconceivable 15 years ago.

3; DEVELOPMENT IN PARAMETERS OF HULL FORMS Up to this date stability is assessed according to the conventional method by applying the experi ence gained, with other ships. This is correct, as long as the ships' form parameters are similar. This could  still'  be taken for granted, when IMO Resolu tion A 167 was introduced twenty years ago. Suffi cient knowledge was available respecting the stabili ty limits, at least of the majority of ships then existing. However, with progressing containerisation the principal dimensions of dry cargo ships have changed fundamentally. In the diagrams Figs. 1 and 2 the mean values of the B/D and L/B ratios of all dry car go ships at present classed with Germanischer Lloyd have been plotted over their respective years of   con struction.  struction.  For the period from 1960 to 1985 the dia grams show a continuous increase in the B/D ratio from 1.69 to 1.96 and a decrease in the L/B ratio from 6.9 to approx. 6.2. This means that on an ave rage in the coursé of . the past few years the breadths of ships have increased continuously. (The remarkable increase of the ratio L/B in 1985 may be incidental. This ratio varies between 1980 and 1985 as follo fol lows: ws: 1980: 1980: 6.19; 1981: 6.45; 1982: 1982: 6.26; 6.03; 1984: 1984: 6.13; 1985: 1985: 6.35, as plotte plo ttedd in 1983: lines.)) Fig. 2 wi th dotted lines.

- 125 -

Bi D

Thus,  Thus,  as sis sis

of

coll ect ive

changed

2,00

i—i

as

1,90

- - -

1,80

• ;

•

I

• ••

:

i

:: •• :' •

1,70 1,60  fig. I

ship

being

ever,

i i

e mpir ical

as

on

the

sta bil ity

is one

fitability

a

is

or

at

the

decis ive

reliable

dimensioning

greater

risk

it

to

is

How

utilization for

the

logical

that

than

of pr o

not

criteria

stability

to

regarded

incomplet e.

aspects

ship,

ba 

owing be

optimum

container a

to

least

hand

on 'the

which

meanwhile

other

of

judged

values,

have

questionable

constitute

1 1 1

st abil ity

designs

sufficiently

t

befo re,

today was

the

case previously at the time of introduction of Reso lution A 167. Against this background in the Federal Republic

1960- 1965- 1970- 1975-1 1975-1 1980- ^ 196/. 1969 1974 1979 1984 1984 2

of Germany for the first time in 1979 systematic at

Development of the medium ratio B/D of all

teria,

>.lr'y  cargo

for

their

ter

taking

tics

than

tempts were made of the development of stability  c r i 

ships classified with CL, depend

ing on their date of completion

L

which

are

specific

different

dimensions

into has

for

account

been

defined

and

their

possible

ships,

i.e.

thus

bet

shapes,

stability

with

the

characteris criteria   avai criteria

lable so far, which are essentially those of IMO Re

B 7,00

solution A 167, is being applied worldwide up to the present day to all sizes, shapes and types of ships, although

originally

tain

category

IMO

weather

having

been

provided

ships

only.

Later,

of

criterion

was

for the

introduced,

a

cer

so-called

the

develop

ment of which was completed recently. However, this

6,50

weather the

wording

criteria to

Development dry

cargo

of

the

ships ships

medium

classif ied

ratio with

of

that

A

562

(1*))

Resolution,

must,

be

subject

applied

to

to

ships

L/B GL ,

of

of

other

A

167

and

criteria

to

larger

recognized

ships

supplementary

the

Administration)

by

these are, however, not defined or rather not exist

1960- 1965- 1970- 1975- 1980- in 1964 1969 1974 1979 1984 CO

Pig. 2

(Res.

of up to 100 m in length only supplementary to the

L-,

6,00

criterion

ing.  The

disadvantage

of

both

methods

is

that

they

do not adequately take into account the existing  dif

all

depen d-

ferences proved

in

stability

during

tank

characteristics

tests

-

the

and

-

resultant

as

different

stability requirements.

in« on their date of completion

i.  i.  "DYNAMIC" OR "STATIC" METHODS 7 ». REPERCUSSION ON STABILITY It As

a

numerous signs;

ships

this

at

trie

in

general

by

far

in

many

of

e.g.

day

the

the

ships'

stability

is

as

against

while

though

no

stability

assessed

a

against

ception

from

cargo

ships

water

to

container

ships

Rahola's

a

this

capsizing,

according

stability

to

the

the

findings

curve

so

decrease

mits

very

likely

pected.  pected. 

ship

moments,

indirectly

cater

in

but for

water calm

already the

ship

a

ship

could all

complying

with

sustain

all

forces

attempts

to

directly

these

li

be

ex

to

calculate

criteria

a lot, but even if we suppose that we were able to ship

irregular

seaway,

simply

a

still

the dynami c behavi our o f a. ship in a seaway t el l us us

particular

stability,

of

the

to

still

nore

of

ability

heeling

did

that

Surely,

exactly

range

course,

in a seaway - simply by setting the "safe limit" for

employed so far, which, however, almost entirely ig the

Of

physical

counteract

this

effect

onset.

the

is

resultant

crite

well

dry

between

stability

take dynamic effects into account. This is a miscon shows

experience

that

de

curve

of

argument

ria using the still water righting arm curve do not

sixties

range

ongoing

of

early

reliable

and

an

the

modern

connection safety

of

reach

older

particularly

During

range

hardly

of

characteristics

demonstrated

60°,

Even

range

changed

stability

instances on

stability

stability.

exceeded

the

upon

be

of

the

50°.

the

have

can

range

available in

result

is

because

this

had never been a problem.

calculate

- 126  /

• 

the

to we

intact

just

stability

successfully

would

always

level meet

have

for a

to

a

given process

the result in  in   order to  to   cover also a l l othe r possible irregular seaway conditions, combinations with wind forces,  forces,  e t c . .  This means adding a  margin or  or  multiply ing in g by a  safety factor. Looking a t  this, one  one   easily realizes that again th e  physically exact -calculation stops a t a  certain point  point  and the  the  "safe values"  values"  are only established by  by  doing a  further step which  which   has t o be  be  characterized as one of  of   estimation and  and   summari zing.- Th e  differences  differences  in all the approaches trie d ou t in  in  recent years in  in   substance consist in  in   this g e  neralizing estimation step done a t  different points. All methods proposed up to now - and  and   certainly also in  in  future -  stop th e true physical approach soo  ner ne r or  or  later but at any rat e b efore t h e  final appli cable value is  is   formulated. These thoughts automatically make i t  evident, that there  there  is no  no  justification fo r  dividing the va  rious methods of  of  assessing safe intact stability v a  lues into dynamic and stati c ones. Starting from this point one can  can   easily see se e  that a  stability   c r i  terion developed  developed  on the  tests   in an the  basts of  of   model tests  irregular seaway is as  good  as any  as  good  any  theoretical  c a l  culation approach - i t  even is  is  believed to be a su  perior  perior  way,  almost   ali_  dyna   dyna way,  because in  in  model tests almost  mic influences acting on a  ship  ship  in the  the  seaway cho sen  sen   are. implicitly covere d, a  degree of  of  correctness which will hardly even be  be  achieved by  by  calculation. The estimation steps  steps  in the model test approa ch is the establishment of a  correlation between  between  the exact safe intact stability values  values   for one  one  seaway (or (o r at  at   best a  limited number of  of   seaways) to al l the possible seaways a  ship might encounter in it s life. How  How  this correlation is  is  done is  is  explained in another context.

» length according  to  LLC 66 ' B  a  moulded breadth   0'. a  moulded depth, corrected   for •  hatches  or   trunks   KG   m vertical centre of gravity Cß » block coefficient C#  a waterplane coefficient

where L

where 0  h  b  2I/y

c * KG i

I d

CB

~&

y  KG

C W  W 

d and L > 100 m

1100  '

V

L

2  - £ I H

2D-B 

0' * 0 • h •

0

= = a *

8 — 2

6>

L

moulded depth [m] height of hatches [m] breadth of hatches [m] sum   of length of hatches within a range of  L/4  from JH   [m] (see sketch Flg. J)

*I

LL !

rr^

r

6. FORM FAC TOR  TOR   "C"

d  ' D'

[m] [m]

The imaginary moulded depth 0' is determined in ac cordance with the following formula:

Fig.

Idea, approach and  and   course of  of   action  action  of the  the   G e r  man series of  of   tank tests and  and  their evaluation  evaluation   was introduced and commen ted upon on other occasions (see bibliography). A t  this time only certain a s  pects a r e  spoken to  which  are of particular int e to   which  rest  rest  for the considerations here o ffe re d. In a  formula proposed  proposed  by Dr.  Dr.  Blume (Hamburg Ship Model Basin)  Basin)  for a  a  form factor "c" the  the  f o r m , p a  rameters essentially determining a  vessel's stabili t y ,  as has  has  been observed and an d  proved by  by  measurements during t he  he   tanks tests conducted in irregular sea ways,  ways,  have been compiled in  in  accordance with their respective significance. Th e  formula reads as  fol lows:

[m] [m]

r

|—'« —l

I

'M '

1  ~\f— 

ymiHTm  \/>À>>»»»»mi -v i  ymiHTm 

•

•

' K 

!

i r — y

P

3 Symbols, used in the formula for Formfactor "c"

Th e new stab ility requirements using this "c" -fac tor for fo r a  particular ship  ship  and its  its  actual loading condi tion result from th e followi ng formulae :

GZJCP

>

GZmax

 c 0.02

 E 400

EjOO-400 127 -

c 0.05  c 0.012

EXP

to

0.04

>

 c 0.035

c

0.007

G

 [m]  [m]  [m • rad]  [m - rad]  [m • rad]  [m • rad]

whore GZyfi V C^nax C^nax  Ejgo  EjCP  £Q £j0O-400

-righting lever at 30P Inclination *  maximum righting lever «  area under GZ-curve GZ-curve up to XP •" area under GZ-curve up to  40° * total area under GZ-curve - area under GZ-curve between XP  and 40°

Except for. the criteria Eo and GZ ma x , these are the assessment criteria known from Resolution A 167, however, with "weighted" "weighted" and thereby . variable mini minimu mum m values val ues.. - ' 7. TRAN TRANSFOR SFORMATI MATION ON OF TANK TANK TEST RESULTS INTO LIMITING VALUES Now, Now, how how did these* t hese* Criteria come about? On the basis of the tank experiments experiments it was possible to fairly accurately ascertain for the individual ships investigated investig ated the limit bêtwénn "safe" and "unsafe" - namely, in following or quartering seas. However, as the resultant stability requirements ap peared to be unrealistically stringent, - a natural result of the extreme characteristic of the seaway chosen, - a way had to be found for reasonably har monizing the results obtained by experiments and practical experience. To this effect, the stability particulars of nearly 150 container ships were ana lyzed and the most unfavourable loading condition listed in the individual stability booklets was ca tegorized into three grades of stability, i.e. ex cellent, normal and limiting condition. The statis tical analysis of the values thus obtained led to the criteria already mentioned, which so to speak' represent the category "limiting condition of stabi lity" i.e. just acceptable cases. As already men tioned previously, these values are markedly below the requirements directly deduced from the experi ments. This will be surprising to unprejudiced rea ders.  However, it has to be considered that the tank tests were carried out under comparatively ex tremely severe seaway conditions, and it is easy to understand that in view of a different, less severe seaway less stringent stability requirements would have been imposed with regard to safety against cap sizing. Even ' more more important, during during the tank test te st

oz

"V

^ i

o-

Fig. *

8. PHILOSOPHY PHILOSOPHY OF THE THE NEW NEW METH METHOD OD Stability criteria presently used - be they in ternationally accepted or set forth by national Admi nistrations in addition to or in lieu of those recom mended by IMO - distinguish the ships by length and sometimes by types. This tacitly implies that the hull forms and all other features affecting stabili ty (superstructures, hatch coaming, etc.) only have an influence on the safety against capsizing to an extent as reflected refl ected in the static stat ic righting righting ar arm curve. curve. This This simplification simplification is certainly, acceptable as long long as the general parameters of the hull forms, such as L/B, B/D ratios etc. vary within certain limits so that the dynamic behaviour of the hulls in a seaway can be taken as having similar characteristics. Looking at the evolution of hull parameters as mentioned within the last 20 years, i.e. since the mid-sixties, when the traditional general dry cargo ship started to gradually turn into a "wrapper for containers", one has to admit that the above assump tion regarding the physical similarity of the dyna mic behaviour cannot be upheld. This is already ob vious at least in its tendency, by comparing only the curves of static stability of a modern cargo (container) ship with its predecessors. (Fig. 4)

A

GZ (ml

(m|

0,20

the most critical situation was intentionally, searched for while in practice, as a rule exactly the opposite will occur, furthermore, no "nauticals" counteraction against dangerous situations were ta ken. When dimensioning a vessel's stability, it will hardly be possible or at least extremely difficult to determine the seaway to be assumed, the probabili ty of occurrence of this seaway under unfavourable loading conditions, and the practical qualifications of the crew, to logically relate these aspects to each other ' and and to draw draw conclusions as to a vessel's required stability. However, when relying upon the statistics of a small fleet sailing successfully, the parameters mentioned are considered almost ex haustively, so that the stability criteria, as weigh ted by form form factor fact or "c" may may be regarded as a good com com  bination of scientific research, practical experi ments and experience acquired.

0,20

 \

0

/ • -

/)

^

 — ! •

80" 0* 20* 20* 40° 40° 60* 60* 80" 80" 0' 20* 40» 60' 60' Comparison of GZ-curves of a traditional dry cargo ship (left) and a modern container ship (right) with identical areas under the GZ-curves.

- 128 -

AU considerations of such complicated physi cal phenomena like the motions of a shaped volume in a seaway necessitate mathematical simplifica tions or practical condensation into formulae or diagrams one can work with. After having closely followed up the attempts for decades to adequately cover ships' movements, by mathematical models, we came to the conclusion that a promising alternative was to use series of model tests, in which the mo dels in real service conditions are subjected to ir regular seaways. Here, in an all-embracing way thé dynamics including all major and minor effects of the form of a ship are covered, even those which do not result from buoyant but from two-dimensional parts like bilge heels and bulwarks. - It can cer tainly be stated that such trials are almost, true copies of reality. The problem at this stage is now to condense all the influences into a number of pa rameters, by which the variation in form between ships are described. This number of parameters ex pressed in a formula can be related to the descrip tion of the stability we are all familiar with - the static righting arm curve in still water. If we now have two ships of different form characteris tics,  we can "weight" the curves of static stabili ty with the factor compiled using certain form cha racteristics and thus compare their safety against capsizing in a certainly much better fashion than by using just the usual static stability curves (see Figs. 5 to 10). This is the basic philosophy behind the propos ed "form-factor c". (In Ref. 1. and 2. a different presentation was used to explain the idea of the form factor; it might be beneficial for the indepth understanding of  . the approach to read the relevant parts of the previous papers again.) 9. APPLICATION OF THE METHOD As already mentioned, application of the stabi lity criteria weighted by form factor "c", derived for the respective ship, leads to greatly varying stability requirements. The D'/B ratio has a parti cularly strong, effect, as the square of the ship's breadth is used in the form factor formula; Accord ing thereto great breadth, low depth ships require a higher stability than ships of equal breadth, but greater moulded depth. This tendency is clearly proved by the results of the tests conducted in the Hamburg Ship Model Basin. In order to offer an idea of the effect of ap plication of the proposed stability criteria on ships in service, for three ships with different B/D ratios still water righting arm curves were cal culated, which just about come up to these crite

ria. The ships chosen have B/D ratios 'of 1.64, 2.0. and 2.43 respectively, Le. a relatively narrow ship,  an extremely broad ship and one in between, all of them are modern container ships. The form fac tors "c" range between 0.186 and 0.083. Presumably the "c" values of most of the ships nowadays in ser vice will range within these limits, so that the examples selected offer a clear idea of the possible effects of application of the weighted stability cri teria. In order to enable a direct comparison to be made, the still water righting  ' arm curves required according to Res. A 167 were included in the GZ-diagrams. For each of the three ships taken as examples two draughts were investigated. The results are shown in Figs. 5 to 10. 10.  MINIMUM GZ-CURVES ACCORDING TO GERMAN APPROACH (• ) AND RES. A 167 ( )

Vessel I: L = 161.70 m B/D = 1.64

B = 23.40 m

D = 13.50 m

L/B = 6.37

GZ

[ml [

1 

0,40 

J~l

7^^

0,20 • 0  Ue=L-  0°

Fig. 5

L_ 20°

J

—L 40°

Vessel I d = 9.95 m form factor "c" = 0.1861

1 \ I1 60°

GZ [m]

0,80 ——i

1

1 / T \

i—I

20c

40°

60°

0,60 0,40 0,20 0 Fig. 6

-129 -

0°

Vessel I d = 8.50 m form factor "c" = 0.1507

Vessel fie

Vessel Dfc

L *  125.«0 m S/O  * ZOO 

O s W» m

B h  ISJSO m

L s 107.27 m B = 19.60 m B/D  » 2.W L/B =  5.47

l A > 6.97

D  = SU»

GZ (m] 0,80

j -r

0,60 0,40 ^--

0,20 ff  •*

***"

•

1/'t

1

0« Fig. 7

20»

40°

60°

0° Fig. 9

Vessel ü

d s 7.00 m form factor "c" = 0.1573

20°

1

>

V

 \

'vi

1 * 1

40

 \

Vessel III d = 6.15 m form factor "c" = 0.1092

GZ (m] 0,80 h GZ

(m] 0,60

:—^S

0,40

/

,

0,20

\A 8

0 Flg. 8

/  >

y \

' is '

s

V A

•N

0,60 h 0,40 h

v

-H

 \ '

•

20°

y- 

•

• 

A V  \ f '—:—'

40°

Vessel II d = 5.00 m form factor "c " = 0.09*9

0,20

\ 60° Fig. 10

- 130 -

Vessel HI d = 5.00 m form factor "c" i 0.0827

 \

'60°

When comparing these still water righting arm curves, it will be noted that (or ship No. I appli cation of the proposed weighted criteria does not 'lead to more stringent requirements than those based on the criteria of Res. A 167. On the contra ry, with the smaller draught even lower stability values would have been possible, which would, how ever, have resulted in a lesser initial stability. It may be interesting to. note in this context that as an individual criterion the initial metacentric height GMo does not permit any conclusion to be drawn as to a vessel's safety against capsizing. Therefore, the proposed set of weighted criteria does not contain any GMo requirement. On the other hand,  it is of course evident that for operational reasons GMo must not be ton small, SO that it appears to be reasonable to <>r least adhere to GMo •* 0.15 m as required in Res. A 167, even when . applying the new criteria. With ship No. II at a draught of 7.0 m the still water righting arm curve weighted with the "c"  factor is only slightly above the curve requir ed according to Res. A 167. In the weighted curve the criteria GZ300, G2 m a x ,  E(,no, £300-40°, and Eo all are determining at the same time. At the draught of 5.0 m the weighted lever arm curve is markedly above the A 167 curve. It is de termined by the criteria E  300 and Eo. With ship No. Ill the still water righting arm curves required according to the weighted criteria are most noticeably above the curves required accor ding to Res.A 167. In this case the weighted curves are determined by the criterion Eo exclusively. As a result of this, on ships having traditio nal maindimensions, as were common at the time of introduction of Res. A 167, even in the event of ap plication of the weighted criteria no more strin gent stability requirements are imposed than previ ously. It is hereby proved that for such ships the value of Res. A 167 is undisputed. This aspect is very important when considering the introduction of such a new method, which is supplementary to the set of criteria used so far. For ships having principal dimensions deviat ing from these ' traditional ones, i.e. essentially for ships with an increasing B/D ratio, when apply ing the critera weighted with the form factor "c", more stringent stability requirements will result. In the light of the experience gained by the tank test this is indeed necessary. If these new crite ria are observed, the safety standard of such ships is approximately equal to that of "traditional" ships,  or vice versa: if such ships are assessed in accordance with the criteria of Res. A 167, their safety standard can be considerably lower. For ship

No.  HI it has with lever arm requirements of capsizing would ing seas.

e.g. to be taken for 'granted that curves corresponding to the minimum Res. A 167 a considerable risk of exist in heavy following or Quarter

11 .  WHY NO ACCIDENTS SO FAR It can be argued that up to this date hardly any capsizing accidents of container ships have been reported on. This is true also for extremely broad ships, although surely in many instances their stabi lity is definitely considerably inferior to that re quired according to • the weighted criteria. Thus, practice appears to disprove the risk of stability ol such ships, as reckoned with in the preceding statements. In this context it should be mentioned that the tank tests were carried out with models not carrying deck containers. Had this been the case, in many of the runs performed capsizing would definite ly not have occured. During the indifferent phase, which could often be observed prior to capsizing or to uprighting, the stability behaviour of the models would have been essentially improved by additional buoyancy (= containers) in the midship area. It is logical to presume that the same applies to the full scale ships, i.e. that with their buoyan cy the deck containers contribute to stability in critical moments, thus frequently preventing capsiz ing.  On the other hand, it would be more than dubi ous in considering safety against capsizing to make the survival chances of ships dependent on thin-wall ed,  possibly inadequately lashed and perhaps slight ly rotten • tin boxes. The risk implied would become incalculable, so that the supporting effect of the containers should be kept as a silent reserve only. 12.  FINAL REMARKS

As compared with previous methods, the method described above for assessment of the stability of modern container ships means a considerable prog ress,  as it enables stability to be assessed from case to case, so that the different behaviour at sea of different ship designs is taken into account pro perly. So far, the applicability of this method is   con fined to a certain category of ships, which should have the following characteristics: . continuous deck with hatches and normal-type

- 131 -

forecastle. Deckhouse arranged' aft on a poop or directly on the weather deck. Vertical shell in midship area above the waterline.

If these characteristics prevail, we now have a tool "ready to use" - a feature which compares favou rably with all other approaches to reach improved stability criteria. The difference is that those others - might they be as complete in their theoreti cal consideration as they will - as a final step to practical use still require the fixing of actual li miting values in a very global way. This final step normally has no direct link to the method offered, but is done relying on "experience" or very general 'considerations. It is not the intention of this paper to downrate any of the other approaches, previously and presently proposed. It is beyond doubts that eventu ally comprehensive calculation and assessment me thods will result, which give the future naval ar chitect the possibility of calculating ships' beha viour in a seaway as others can calculate the ef fect of laminar flow of air on a   foil.  But until this stage is reached, the "c" -f acto r method can be used successfully.

REFERENCES 1.

Report on Stab ilit y and Safety against Capsiz ing of Modern Ship Design, International Mariti me Organization, Paper SLF 3<>, submitted by the Federal Republic of Germany

2.

P. Blume, H.G. Hattendor ffj An Investigation on Intactstability of Fast Cargo Liners, Second International Conference on Stability of Ships and Ocean Vehicles. Tokyo, October, 1982

3.

H. Hormann; Judgement of Stability - Questions to be Solved - A Contribution from the Point of View of an Approving Authority, Second Interna tional Conference on Stability of Ships and Ocean Vehicles, Tokyo, October, 1982

<». 

O. Krappinger, H. Hormann; Kentersicherheit, Problemstellung und Lösungsansätze, Yearbook of "Schiftbautechtiische Gesellschaft". 78th Volume 198«

- 132 -

Third International Conference on Stability of Ships and Ocean Vehicles, Gdaris£, Sept.   1986

 [0)\o) Paper 3«3

THE APPLICATION OF SHIP STABILITY CRITERIA BASED ON ENERGY BALANCE C . K u o , D .   V a s a a l o s ,  «J.G. A l e x a n d e r , D *  Barrie  ABSTRACT This paper gives a  brief description of the

Thé initial research work, from  1973 to 1976,

Strathclydc approach to developing intact stability

concentrated  on an  entirely theoretical approach

criteria for ships and   outlines the  progress made

 because it  offered solutions "superior" to the sta

since the International Conference on the Stability

 bility criteria based  on empirical methods,  see

of Ships and   Ocean Vehicles in Tokyo in 1982. The

Ref.(t).  Many of the  research techniques proposed

 practical application of the criteria is highligh

for assessing ship stability initiated during that

ted, along with the assessment of  their applicabi

 period, such as Lyapunov methods, are  still being

lity through comparisons with available stability data and   experience. 1.

INTRODUCTION

used.

The main lesson, however, has been that the development of stability criteria which take full account of vessel dynamics in a realistic environ  ment is a  long-term process. It was  suggested  at

 A number of  basic requirements must be met

the first International Conference or  Ship Stabi

when designing a  ship for operation in various sea

lity in 1975 that the task would need "another hun

ways, but one of the most fundamental is undoubted

dred years! ".

That estimate may not be  accurate

 but it does reflect the magnitude of the problem.

ly satisfactory stability. It is generally acknowledged that a  ship must

Consequently, during the period 1976-79 the empha

 be "stable"  in order to perform  its function effec

sis was placed  on developing sound  and   usable  cri

tively, but an  accurate definition of  this concept

teria which could  be applied  at as  many stages of

has yet to appear.

Furthermore, once a  criterion

the ship design process as  possible. This revealed

for judging stability has been developed, questions

that for a  complex subject like ship stability it

arise about how far it is  valid  for all  loading

is essential to devise a  strategy for tackling the

conditions and   seaways;

how its requirements can  be  met in  practice; whether its application will inhibit a  vessel's performance, and so on. The real crux of the problem  is the need  to quantify the "stability" of a  ship, in a way that represents reality acceptably and is, at the  same time, in a  form that designers  can use. The prob lem    is not new. It has been receiving attention fcr over a hundred years . However, despite  its  being the focus of considerable attention in recent years progress has been very limited.

 problem  in carefully-planned stages. The resulting "levels of stability" approach was directed  at re lating the reliability of a stability assessment to the quality and   quantity of input information avai lable and the current state of knowledge, see Ref. (2).

In addition, some of the elementary levels of stability were developed  to demonstrate the appli cability of the approach. During the period 1981-85, as  part of the SAFESHIP project sponsored, by the UK  Department of

There are

Transport, a  method  was  developed  for  assessing

 many reasons for  this state of affairs, key ones

ship stability criteria that takes explicit account

including the  difficulty  of the  problem  and an

of the effects of wind, waves and    vessel motions.

incorrect emphasis of research efforts.

Criteria were proposed, based  on an  energy balance

This paper will summarise the lessons learned

approach, and    refinements have been sought  and

during the  past thirteen years of  research into

introduced  in the light of experience gained from

ship stability, outline the key   steps involved  in

 practical applications.

the development

of  Strathclyde's practical ship stability criteria and   highlight the effort expen

success in  proposing a 'new criterion depends on

ded  in  developing confidence  in the proposed cri

 progress in the co-ordination of  sound theory with

teria.

 model experiments, full scale data and    practical

The key lesson to be borne in mind  is  that

experience, together with the  good fortune every 2.

LESSONS LEARNKD  A number of   research teams at the University of Strathclydc have worked  on  ship stability since 1973 and.it would  be. helpful to  make a brief exami nation of' the  lessons learnt during these thirteen years.

researcher needs.

Concentrating on any single one

of the  above areas is  unlikely to  lead  to the desired answer.

13 3 -

3.

INTACT STABILITY BALANCE

CRITERIA

BASED ON

ENERGY

. For a long time now strong support has been voiced at IMO for a physical approach to the deve lopment or ship stability assessment, that is, an approach taking explicit cognisance of the influ ence of such physical phenomena as wind and waves. Over the past ten years or so several proposals for  modifications to the still-water regulations have

can be largely avoided by following certain design and operational guidelines, it was decided to con fine research work to the establishment of stabi lity criteria on the basis of the second capsizing  mode .

following/quartering séas, getting caught on a wav« crest, and "flopping" over with little or no preli  minary rolling. In addition, it is widely acceptmi that following/quartering seas constitute the most

 been made, culminating in the recent proposal by

dangerous situation for all vessel with the excep

IMO (3) of what is commonly referred to as a "weather criterion". However, despite the fact

tion of very small vessels, which can be over whelmed by severe beam seas.

that there is some evidence of success in the application of weather criteria in several coun tries,  doubts have been expressed about their suitability for wider application.

The element of beam wind has also been incor  porated in the situation described above, which as indicated by Arndt et al (6) - is by no means

This is princi

uncommon, even though it seems to be a contradic

 pally because the weather criteria are based on a static treatment and the effect of waves is not

tion in physical terms.

explicitly modelled. In addition, the criterion Is strictly applicable only to relatively small ves sels.

 b)

above, which includes all possible known effects. However, in view of the severe limitations on accu

have been made to IMO in relation to the effect of

racy in measuring even such simple parameters as the position of the centre of gravity, there is no

waves on stability assessment and IMO has confirmed this to be one of the priority topics for attention (4).

virtue in seeking a very precise mathematical iiiod.il. This is the reasoning behind the adoption

The stability criteria developed at the Uni

of the "levels of stability" approach. There is, undoubtedly, a need to agree on a roll equation for

versity of Strathclyde during PHASE I and PHASE II of the SAFESHIP project follow the physical ap

use in stability research but an equation that can accurately predict vessel behaviour right up to the

 proach in that they explicitly model the effect of wind, waves and vessels motions. They are also, as will be demonstrated in the following sections, ap

instant of capsize does not exist. On the basis of the above, and in order to

A brief

 maximise the interim benefits at the earliest pos sible stage, research attention was focussed on the

explanation of the Strathclyde criteria is given next. For further details, see (5).

following equation of ship roll motion which con tains terms that can be readily computed and are

The; Basic Approach of' the Strathclyde Criteria

considered to be important for theoretical reasons

To derive a quantitative measure of ship sta

and as a result of experience (7).

 bility on the basis of a physical approach 4, it is necessary to start from a physically realisable and  potentially dangerous situation, then model the vessel's behaviour in the said situation mathemati

ïvt

+ C e  + 5 A (SZ  O r V O * * <"•*>

where

(^ $

(l)

is the roll velocity is the roll acceleration  t is the time $ is the roll angle I v is the virtual roll inertia C - is the equivalent linear damping coefficient g is the acceleration due to gravity  A is the displacement <&($£) is the time-varying roll restoring lever computed in regular waves under the free trim condition, see (7)  yj(4>)  is the wind heeling moment.

cally, and finally develop criteria by rationally simplifying the mathematical model. These key steps will now be elaborated in relation to the stability cr.iteria developed at Strathclyde Uni versity. a)

Mathematical Model

 A general form of rolling equation may be used to describe the dangerous situation referred to

But the need for a physical approach was never in question. In particular, several submissions

 plicable to all ship types and sizes.

This mode pertains to a vessel steaming in

Potential Capsize Situation The three generally-recognised capsizing modes

are!

i) low cycle resonance (dynamic mode) ii) pure loss of stability (quasirdynamiG mode) iii) broaching (dynamic   mode). Since the fjrst mode of capsize can be avoided through proper ha.ndling by the crew, and broaching -. 134

c)

Proposing Stability Criteria (Energy Balance

 Method) By analogy with  the weather criteria,  a gene ralisation  of  Moseley's dynamical stability theory is u&ed  to derive a  quantitative measure of sta  bility. Such  a generalisation is necessitated  by the fact that moments now in the equation  of motion depend  on t and as  well as on $ .  Moseley's theory involves  the balance of ener gy between exciting  and restoring moments  and can  be established only for autonomous roll equations such as that pertaining to the weather criteria. For Equation (1) the first integral over  a half roll cycle beginning  at A  will take the form

2  f* £ <£ + / [9A6ZO,f ) + Cef-WC$)]d = 0 (2) .

*

*

•

•

capsize situation  has been to choose each parame ter, where appropriate,  in such a way as to giv« On  this basis   the the most stringent case. following input information  is used: i) Ship Parameters: For the critical loading condition, the roll restoring lever CZ( ,t) is calculated  in regular waves of given length, height and direction. This represents  the restoring capa  bility of the vessel at an arbitrary moment  in time as she rolls in following/quartering seas. ii) dynamic through ultimate

 A.=

.

In this integral it is implicit that  r and qS are related   to • via the actual motion record deter  mined  by the differential equation  and initial con ditions as  well as the time t, at which the half roll cycle begins. One possible way to proceed  is  by numerical evaluation  but this is precisely what stability theories seek to avoid. Instead,  the following plausible approach has been adopted  on the basis that the shape of a  representative ex treme half roll can be  closely approximated  by a sinusoidal curve. Since  the important parameters- to know in each case are the extreme angles, d> and qVj , the frequency  of oscillation, U , and  t t', the chosen function  is:

(0

Roll damping:

 / 

according

to  Ikeda's

iii) Environmental Parameters The environment is modelled explicitly by a steady beam wind  and regular waves, as follows:

Using this function, Equation (2) yields u»

[3 tactf) t Q* - wflM i J t 

calculated  method  (8)

(3)

i

The extreme windward roll angle represents the rolling behaviour and is  calculated of ships according to the weather criteria. The starting time of the extreme half roll is chosen iteratively such that the most unfavourable combination between exciting  and restoring roll moments is ensured. The frequency of  oscillation of the extreme half roll  is taken to  be equal to the encounter fre quency using the vessel's service speed.

$ (t) = j . C^+dV ") + 1 0 4 -dp C«Swf>0

x

Dynamic Properties of the  Ship:  The properties of the  ship are  represented the parameters defining the  potential half roll cycle,  i.e.,

(4)

t,

Three different quantitative measures  of  stability 1 have been proposed based  on Equation (4) , see  (5). The easiest to compute is the net area analogy  of the weather criteria which  is equivalent to saying that r^,4> a ) > 0 for 4 , being the least of 50°, the angle  of second intercept, or the down flooding angle. The pictorial presentation  may also be pre sented  in a similar form  as illustrated  in Fig. 1.

 Wind heeling: calculated according to  weather criteria (without gusting) with  a  modification to take into account the variation in the  projected area with angle  of heel.  Wave Direction: following  or quartering  Wave Length: the projected wave length on the ship's centreplane is taken to be equal to the ship's length.  Wave Height: calculated from formulae, see Fig. 2.

Input Information The integral in Equation (4) represents  the net work done by all the moments considered  as acting on the ship, which in turn depends on the  parameters used  to define the ship, the environ  ment, and the ship's roll motion. The strategy adopted  in searching  for possible

- 135 -

empirical

, 4.

APPLICATIONS

OF

STRATHCLYDE

CRITERIA

IN

first four test vessels. The details of HSVA-A are given in Table III.

PRACTICE

The investigation demonstrated very clearly

The evaluation and interpretation of theore. tically-derived

stability

criteria, at

whatever

level of sophistication, inevitably pose the prob lem, of how to develop confidence in these criteria.

((13) for further  details).

that

the

Strathclyde

assessment

method

is,

in

general, no more restrictive than either the IMO  A. 167 or IMO weather criterion.

In all the cases

. During the past five years considerable effort has

where differences have been observed, however, they

 been directed towards achieving such confidence in

have been due to thé fact that the simpler IMO

the criteria proposed by Strathclyde, and the me

 methods did not model features that are known to have a strong influence on stability.

thods used are summarised as follows.

A sample of

results from this investigation, illustrating the a)

above observations, is presented in Figs. 5 and 6.

Assessment of a Large Number of Ships The Strathclyde criteria have now been applied

to thirty-five ships from a wide variety of types

c)

varying in length from ten to two hundred metres.

Correlation Studies with Model Experiments  A good opportunity to test the Strathclyde

It should be noted that so far results have been in

criteria witn container vessels of lengths from

agreement with operational experience and casualty

approximately 135 to 200 metres arose through col

records.

the

laboration with the Federal Republic of Germany.

criteria has been found to reveal discriminative

The Strathclyde research team was asked to compare

features between ships not revealed by the simpler

its criteria with the FRG-proposed criterion, which

IMO weather and statistical criteria.

is experimentally derived (14) .

Furthermore,

the application

of

This is due

Recognising this

to the fact that the Strathclyde criteria explicit

as a further test for consolidating the effective

ly consider the effects of the environment and the

ness of the Strathclyde criteria, the team agreed

vessel dynamics.

to carry out these correlation studies using twenty

This point is elaborated in the

next section.

different test cases. The FRG team have approached the development

Comparison with Existing Criteria

of their criteria by carrying out systematic model

In relation to the search for improved stabi-

tests in extreme seas - to establish limits against

standards it is relatively easy to. propose a

capsizing and measures of safety (limiting values)

new criterion, based on a limited number of appli

and by calibrating the latter through comparison

cations, that would appear to meet the need.

with experience.

 b) i  lity

It

Recognising that the Strathclyde

is,  however, a quite different and altogether more

criteria are not based on a fully dynamic model of

difficult task to demonstrate conclusively that any

vessel behaviour, it was thought that correlation

new criterion is an improvement on its predeces

studies should be aimed at achieving the following

sors, in that it gives a better description of the

objectives: i) To compare the trends of the limiting values

factors that are important for stability. Recent research work at Strathclyde has fo-

established through the model experiments with

;ussed on comparing the developed Strathclyde cri-

the corresponding trends established using the

ieria. with the IMO A.167 and IMO weather criteria

Strathclyde method,

(13).

The features present in these three methods

ii)

of assessment have been identified and their in

To compare the levels of safety indicated by the two approaches.

fluence on the assessment results has been inves

The relevant information for the test vessels is

tigated.

given in Table III.

This has been done

by

systematically

(See (14) for more.informa

varying the ship design parameters and environmen

tion).

tal conditions that influence the effect of each

range of seaways was used.

assessment feature, using the following data:

ready alternative to varying the site of the test

 A:

Fishing Vessel (L = 21.40 m )

B:

Tug/Supply Boat (L = 52.60 m)

C:

Fishing vessel (L = 56.85 m)

D: HSVA-A:

The

 A sample of the results pertaining to the  minimum

Container Vessel (L = 135 m ) . influence

of

each

parameter

assessment results was then considered.

This was employed as a

vessels. .

. Fishery Protection Vessel (L • 64.00)

relative

It will be noted from Table III that a

requirements

(limiting

values) for the

Strathclyde, FRO, and IMO A. 167 criteria and to Seaway I, is given in graphical form in Figure 7,

on

where comparisons can be made of both the trends

Table I

and the levels of safety revealed by the various

illustrates the way in which the parameters that

criteria and the model experiments.

were investigated arose from the various assessment

From these results it will be noted that the

features and Table II gives the details of the

experimentally-derived FRG criteria and the theo-

36 -

5.

retically-based Strathclyde criteria are in close agreement in all cases äs regards the level of

remarks can be made: a) Over the past five years, in association with the SAFESHIP Project, Strathclyde University has

, safety and in most cases they indicate similar trends. Disagreement in trends has been observed • in cases with extreme breadth/draught ratios and this has been attributed to overestimation of dam

developed a procedure for assessing the intact sta. bility of ships, that explicitly incorporates the effects of wind, waves asnd vessel motions and is

 ping in the Strathclyde criteria for high breadth/ draught ratios and large roll amplitudes. It will also be noted that the results of both the Strath

 based on energy balance. The proposed stability criteria have now been applied to thirty-five   ves sels and comparisons have been made with empirical

clyde and the FRC criteria are markedly different from those of the IHO A.1Ó7 criterion. .d)

CONCLUDING REMARKS On the basis of the foregoing, the following

data, other stability criteria, and model experi  mental data. In all cases the results derived from the proposed criteria agree closely with the  evi

Parametric Studies

Once sufficient effort has been expended on developing confidence in any newly-proposed   cri teria, there remains one further task. For regu

dence at hand. On the basis of these applications, refinements have been sought and introduced in the criteria.

latory purposes it is important to link stability assessment to design if new ships are to benefit fully. If a design fails to satisfy the regu lations the designer will be asked to improve its

 b) A systematic parametric, investigation has been undertaken, the results of which have indicated that despite the large number of parameters that

stability characteristics while maintaining the competitiveness of the design. For this to be  possible, knowledge is needed of how, and by how

affect the stability assessment of a vessel, the same qualitative trends can be attained by conside ring only a limited number of such parameters.

 much, design and other related paramaters affect stability. In other words, a systematic parametric investigation is warranted. On the basis of the

c)

research at Strathclyde University at which a prac tical "tool" has been developed that offers a sig nificant improvement over existing methods for

information described in Section ( b) , the following groups of parameters were investigated: i) Ship design parameters

1

judging the intact stability of ships. d) As long as ships continue to be lost at sea, the need will exist for a constant reviewing of

it) Dynamic properties of the ship iii) Environmental parameters. This investigation has identified the most influen

stability standards and increased effort to provide  more •effective criteria. As regards the Strath clyde criteria, however, it is felt that the stage

tial parameters from the point of view of stability assessment äs well as the sensitivity of stability to changes in these parameters.

has now been reached at which an improved ship stability criterion - inspired by the UK Department of Transport - could be put forward at an

Of.the ship design parameters investigated, KG (height of centre of  gravity),  L/B (length-to breadth ratio) and absolute size were found to have

international level as a practical contribution towards greater ship safety.

the most influence on. stability assessment. Changes in any of these could affect this assess  ment far more than specific changes in form. The  parameter to which stability is most sensitive is

 ACKNOWLEDGEMENTS The research outlined in this paper was spon sored by the Marine Directorate of the U.K. Depart  ment of Transport as part of the SAFESHIP Project, and their support is gratefully acknowledged. Par ticular thanks are due to Mr H Bird for his inte rest and help, and to those who have kindly made data and experience available to us. Special thanks are also due to Miss C Hutcheon for her help in the preparation of this paper.

KG, a fact which highlights the importance of esti  mating this parameter as accurately as possible. The research findings have also demonstrated very clearly the importance of selecting suitable values for the environmental parameters. A sample of sum  mary results for a small vessel (vessel A) and a large vessel (vessel HSVA-A) is given in Figs. 8 and 9; , The investigations'described above underline

REFERENCES 1. KUO, C and ODABASI, A Y : "Application of Dynamic Systems Approach to Ship and v Ocean Vehicle Stability". Proceedings

the amount of effort needed to validate any newly  proposed criterion, although there are, and always will be, disagreements regarding the conclusiveness

Int. Conf. on Stab, of Ships & Ocean •Vehicles, Glasgow, 1975-

of such tests. •

'

•

/

.

'

.

'

The stage has been reached in the stability

.

'

'

.

• '

- 137

2. - - -  Intact Stability Research, Ninth Work shop: Ross Priory. Dept of Ship &  Marine Technology, Univ. of Strathclyde, September 1979« 3. _ _ _ IMO RESOLUTION A.562(14) Recommendation on a Severe Wind  and Rolling Criterion (Weather Criterion) for the Intact Stability of Passenger and Cargo Ships of 24 metres in Length and   Over.  A 14/Res.562. 16th January 1986. 4. - - - IMO Publication SLF 30/WP/7, Report of the ad hoc Working Group on Ship Sta  bility, February 1985- "  5. VASSALOS, D: "A Critical Look into the Deve lopment of Ship Stability Criteria Based  on Work/Energy Balance". RINA  W4 098 5) ,  issued  for written discus sion.

in Japan", Trans. INA,  Spring Meeting,

1959. n #

1985. 12 .  ANDREWS, K S. DACUNHA, N M C and   HOGBEN, N: "SAFESHIP: Environmental Aspects", NMI Report No. 185. ^. BARRIE, D:   "The Influences of Ship  and Environmental Parameters on  Stability  Assessment", Ph.D. Thesis, 1986. 14. - - - I M O Publication SLF/34: "Report on Stability and Safety Against Capsizing of Modern Ship Design", September 1984-

6. ARNDT, B, et al: "Twenty Years of Experience -  Stability Regulations of the West German Navy". 2nd Int. Conf. on  the Stability of Ships and Ocean Vehicles, Tokyo, 1982. 7.  MARTIN, J, KUO, C WELAÏA, Y: "Ship Stability Criteria Based  on Time-varying Roll Restoring Moments", 2nd Int. Conf. on the Stability of  Ships and   Ocean  Vehicles, Tokyo, 1982. 8. IKEDA, U, et al: "A Prediction Method  for Ship Roll Damping", Dept of N.  Architecture, Univ. of Osaka Prefec ture, December 1978 9. BARRIE, D: "Incorporating Wind Heeling • '.-.. Moments in Stability Criteria", Internal Report, Dept of Ship and   Marine Technology, Univ. of  Strathclyde, February 1985 10.  YAMAGATA, M:  "Standard  of Stability Adopted

TABLE I !

 ALEXANDER, J G: "Equivalent 'Design Waves' for Stability Assessment", Int. Report, Dept of Ship 4 Marine Technology, May

 AUTHORS' DETAILS Prof Chengi Kuo.BSc, PhD: Head  of The Department of Ship & Marine Tech nology University of Strathclyde GLASCOW, Scotland, UK Lecturer in the Depart  ment of Ship 4  Marine Techology

D Vassalos. BSc,  PhD:

University of Strath clyde GLASGOW, Scotland, UK Research Assistant in the Department of Ship & Marine Technology University of Strath clyde, GLASGOW, Scotland, UK Consultant  (JO) Y-ARD Limited Charing Cross Tower GLASGOW, Scotland, UK

J G Alexander, BSc: "

" '

'

 Barrie BSc: D A 

ASSESS MENT FEATURES AMD ASSOC IATED PARAMETERS DETAILS OF SHIPS IH BASIS COUP.TION

FEATUffi

SUB •  FIATORES

Roll Restoring

Roll Darcplng

Windward Boll Angle

(

Position Measured Proa upright,Intersection )

Roll Cycle

Wind Keeling

Kovea'

PARAMETERS

ASSESSMENT METHOD

Design parameters (  Sit»,  Dimensions Linos,  Loading )

Statistical Criteria IHO Weather Cri ter ion Strathclyde Criteria

Design Parameters

S t ra t hc l yd e c r i t e r i a

Design Parameter*

P h a s i n g o f   Wave and Ro ll Motion Roll period Encounter Period Ship Speed Presence or   gusto velocity - height profil«  Variation in windage areo ' with heeling angle Regular / I rregu lar Undlffractod / Diffracted

Wind Speed Above Water Profile Design Parameters Wevahelght Wavelength Direction

IK) Weather Criterion Stratholyde Criteria

Strathclyde Criteria

Vessel A

0

C

11

21.40 6.71

52.60 12.20

56.85 12.19

64-00 11.60

Draught/a

3-35 2-37

S.50 3.40

7.77 4.25

4.49

Displacement /tonnes L/B '

160

1290

3.20

1500 4.66

1.34

4.31 1.62

B

0.472

0.575

VP CM/»

0.761 0.760 0.621 0.620 0.768

KO/.

2.58

Ungth/a Oreadth/n Depth/a

0/ t C

V M V C

IW Weather Criterion

-  Strathclyde Criteria

Strathclyde Criteria

-  138

C

V " LCC/a

.

2.45 -0.77

0.743 0.955 0.602

1.83 O.498 O.772 O.828 0.602

O.773 I.I60

-O.646

4.66 5.26 O.08

5-57 5.37 -1.11

O.525

7-32

1532 5.52 1.63 O.448 0.708 O.836 0.536 Ó.634 O.926

4.63 4-86 -1-53

TAIILK  I I I :

Ship

Sea way

1

HSVAA I 11

r.

AI

9.2

A2 A3

t

A4

n

ui

AS

ir

in

Bl 02

HSVA- l B i 1

IISVA-

Code Vel. No. (ms )

B3 B4

« " " it

n

CI

w

i i

C2 C3 C4

it

ni

CS C6

H

ii

Dl

M

m

HSVA. i D i

i lit

M «3 »4 «5

•

M

" " "

BA

TUST VESSKL UIADINC CON«ITIONS

L (n)

B M

D (•)

UX M

Dlsp tonnes)

T (•) 8.2

2.8

135 0 23.0

10.7

-I.IO8

17620

3.43 4.42 2.8

135-0 23-0 I3S.0  23-0 154-3 26.3

10.7 10.7 12.2

-0:743 -0.717 -1.265

2.8

192.9 32.9

15-1

-1577

13879 6.7 10439 5.» 263OI 9 4 5137O 11.7

3-58 3-58

202.4  32.2

18.85 -5.218

35S52

141.7 22-5 141-7 22-5 1417 22.5

13.» -4-251 132 -3.652 132 -3.278

12194

IS8.0 28.0

16.1 14 1 Ml 14.1 20.1 20.1

33087 22166 17189 12681

2.93 4 54 2-SS 2.SS 311 40 2-55 311

138;3 138.3 138.3 197-5 197-5

3-58

161.0 32.2

2.93 3.58 4-60 3-58

24.S 245 24-5 35-0 35-0

-3.793 -3-319 -2.611 -2.066 -4.741 -3.730

13-8 -0.934 140.9 28.18 12.08 -1.521 140.9 28.18 12.08 -O.7S0 140.9 28.18 12.08 -O.398 201.3  4030 .17-30 •1.073

15873 9007

55&S4

5.86 5.86 5.86

I.? 1-7

(»)

5.99 6.19

9.0

6.23

1.10

6.3 7.7 •4.97

591 5.91 S.91

1-37 1.09 2.07

11.0 9.63 7.88

6.02

0.55

$.38 5.88

6.13

5.88 6.21 6.21

0.53 O.48 0.70

13.75 SOI 13 li.25

19089 14042

CM

2.35 1-94 2.07

64623

28496 24686

Vtve Heljht (•)

9-0

6.03

9.63 7.88

5-9 59 59 6.23

6.13 11.25

0.563 0.188

2.75 2.54 2.50 3-54 2.75

k I  («.1« • 0.1»» ) 0.40'VI

X / (to.O » O.OJM St.«thrlyd« H«lh

'Fi.g.1 Energy Balance Diagram  of  Strathclyde Criteria  ("Butterfly   Diagram").

Fig. 2 Wave Height Determination in Strath clyde Method.

- 139 -

Stiathclyde Criteria iMO Weather Cr ite rion 25

3ig Vessel HSVA-A

Vessel A "

L/Ö : 2. 00

*

L/8 = 3 . 2 0

»

L/B = J. 45

A
15

15

10

10

to

1?

H

16

10

in

20

12

11,

L/B .: 5.60 L/B : 5.87 L/B +6 .3 0

16

18

20

?r>

22 2' XI o" 1

22

O/I

2* 26 1 mo"

O/I

F i g . ] Var ia ti on in Windward Angle (* ,) wit h L/B and n/T

I K !••»

35

Wind Heeling Lever

m

30

/' 25

Vessel D

/

/

/ . . 1 ! >0"1

12

;e

'u

 Wind Heeling Lever

.2

ni

Vessel C Vessel A

ID

/

/... /  »-

15

10

/

/

• ^^'^ .^-"^

^

./'^

8

/

/ .

.

•

•

Vessel D

'

6

(•

IMO lever

5

2

Vessel HSVA-A

-

0

0

10

 Wind speed 'm/s

length  m

Kig.4 Variation of Wind Heeling Lever with Wind Speed  and   Size

,. 1A O -

):•

Kl0 '

20

~N

Net Area xicr 2

Net  Arëà xio" ? 6

'v. 2

\

Vessel

s

20

A

v

Vessel B

15

0, ?«

25

"

••"<.

~'

 \

t.

»«

p

48

5(1

57

KG m

Net  Are» t 

x]0~' Vessel I15VA-A

 —  Str ath dyd e Criteria - • 1M0 Weather Criterion

X 7 T — ~ T S — I f  

W~^&—Jijr

xlO

KG

Fig.5 StabDity Assessment Versus KO

-141

ü'

56

xlO •t

VaHatjbnJn„SjabUU^..Àssesjment with 0/T and L/8 - Vessel A Net Area xlO"2 15

Strntliclyde

1MO WeBthêr C r i te r i o n

Criteria

Net Area xlO 10  \ . N.

X.

O  ...

 .  o/r

Ü/T

iT-Tr7-Ti?r~-n!"~"'~r8"~-w—n

s 1.0

I/O . 1,10 - • I/O 1 ). I0  I'»

. »• »

10

O

P as 19  Form

 Vari atio n In Stability Assessment with  0/1' and   1/Bj; Vessel C

xlO

TMÜ Weather C r i t e r i o n

tStxotliclyde C r i t e r i a

1/9 1 !.».'•

25

 y

20

 /

15

 /

5

Bas ta For«

15

Ö.

0 •

./-"''

O

I/O • * t\

S

10

-?. xlO

l/ B >  J.00

12 

14

16

18

70  

22~~?* xlO-1

0/1

071

0

 xlO  V  -5

  L/B Fig.6  Variation in Stability Assessment with D/T and 

-' I42

xlO"1

xlO IISVA-lt

U

IlSVA-A 12 lü

fi 6 U

'%

ïo"

T3~

TTT

T b

xlO

?ï

"53

5Η "To "

" I n xlO

Ï5

B/l

B/l

,?5

xJO"1

HSVA-C

xlO"

HSVA-D

20

10

16

^

^,-^Ï*.

25

in

âr

q| 25

*» xlü"

B/l

' .10

îT

*S

~Z ?

?0 x l O " ' B/l HSV» Hodtl t' ip tr iu nt i Slrji htly de O it«< ia

r i a . 7 Va ri at io n of' Minimum OZ

wit h U/T

FBG C.i tt ri on . . ._ ?

Bet Ar « ».nul

Length

KO »

 î !6 •

?.a -

30 •

Wind Speed Wawe Height

-

0.5

-

1.0

•

KG •

Lc

2Äi 5

Wind Speeed

W*«  jiti&M

te "

20-

25

u.î

Hot Arc* ta.md

fl 1 6  7  r . r i t f î f i n

l.S •

.

?.o-

3.2 -

31 .e

-

26 -

2.6-

BAsrs d u o

2.583.5 -

32 Q . O O O - - . — 

IS "

7.66

?. eo-

-0.100 J

Fig.8

Influence of Parameter Variation on Stability Assessment .- Vessel A

--

Fig. 9 Influence of Parameter Variation on Stability Assessment - Vessel HSVA-A -

- 143 -

Third International Conférence on Stability of Ships and Ocean Vehicles, Gdath£, Sept.  1986

mm*m> Taper 3.4

THE NORWEGIAN RESEARCH PROJECT STABILITY AND SAFETY FOR   VESSELS IN ROUGH WEATHER

T . Nedréli di- E . Jullumstr«*

 ABSTRACT

1. INTRODUCTION

This paper summarizes some of the work per formed under the Norwegian research program "Sta  bility and   Safety for Vessels In Rough Weather". Future stability criteria should  be  built up around dangerous physical situations. We know that  both design factors as well as human and   weather factors are Important when preventing capsizing In

In Norway we still experience that vessels dissappear or capsize In rough weather situations. For quite some time, stability work and   Investiga tions Into earlier accidents have been performed.

waves. This means that we have tried  to  develope  mathematical models that better describe the cap sizing mechanisms and  extreme responses of a vessel. Better environmental description  are achieved through new analysis of wave statistics and  new wave measurements given from offshore research work. In the  program  we have concentrated  on studying the capsizing mechanism  In a  following wave situation. Results from loss of stability when balanced  on a wave crest are presented. Also theoretical models describing the "broaching to" situation supplied with model test results, are  presented. Theoretical stability is often poorly under stood  by people working at sea. We normally put restrictions on the vessel design, but often, to  prevent capsizing, the actual operational stabi lity is more important. We  have therefore been studying If operational procedures can be put forward more.precisely through operational manuals  a vessel. If this is accepted, operational onboard  stability should  be more integrated into the pro cess of assessing new criteria.

- 145

In the seventies, several Norwegian cargo and fishing vessels were lost due to environmental loads.  This initiated  the research project "Ships in Rough Seas  (SIS). In this project one  focused on breaking waves and how to calculate extreme  motions of vessels in waves, theoretically. The idea was that If one could calculate capsizing events, one might directly put forward rules of restrictions that would Improve the design and the safety.  At the start of the eighties much effort was  put Into the research program called "Stability Criteria". The work was concentrated  on  model testing of vessels in so-called survival tests. From earlier experience and   Investigation One can define several dangerous wave/vessel situations  to capsizing. The idea was to find that might lead  the situation In which the vessel Is most exposed and through the probability of the actual wave to occur, define theSrisk for capsizing. ¥ •

 As critical wave/vessel situations are defined through parameters as heading against the waves, speed  and the actual loading conditions, human decision or seamanship Influence much.  Accepting this, we built up this new research  program (called "Stability and   Safety for Vessels in Rough Weather") In three main subjects:

Ship and cargo Environmental description Stability criteria  We still need to calculate extreme motions in waves to understand the nature of capsizing in order to improve the vessel design.  We still need better knowledge as to the nature of environmental loads (waves, current and wind) and their statistical occurence along the  Norwegian coast.

1.

Traditional design criteria based upon the physical understanding of stability (n critical wave situation.

2.

Risk analysis.

3.

Operational procedures.

This leads to the following sub-projects where some are presented In the next sections (3, 4, 5 and 6) .

He still need to define correct stability cri teria that consider all important aspects when we are to accept or approve the design and/or the operation.

2. THE PROJECT ORGANISATION The research program was started in 1981 as a cooperative project between several Norwegian institutions with MARINTEK being the responsible  part. The project has been supervised by  a commit tee with members from industry, authorities and the research Institutions themselves. Also, repre sentatives from UK have mét in this Norwegian com  mittee. The program has been sponsored by: - The Norwegian Council for Scientific Work (NTNF) - The Norwegian Fishery Technology Council (NFFR) • The Norwegian Maritime Directorate (NMD) Over the years, plans and economy has Changed. Several Items being studied under the umbrella of this program have been worked on Independently, sponsored separately. The basic Idea leading to the plans and the sub-project list, has Its origin In earlier research work and the acceptance that future cri teria should be built up of the three following  parts:

- 146 -

- Developing better theoretical models to get a more precise understanding of the capsizing event. This particular program has concen trated on the following wave situation (see section 3). - Ship motion and Its Influence on the shif ting of cargo. This program Initiated a study on the problem of shifting bulk cargoes. A  more Independent project has been working separately over the last years with comprehen sive studies on the cargo properties as the angle of repose (see section 4). - The original plans included studies on the environmental loads, concentrating on the waves.  However, the program has not directly sponsored such studies due to a change in pro ject economy. Some work being performed with separate financing, mostly from the oil Industry, has been followed closely by this  program (section 5) . - The problem of assessing future stability and safety criteria has been accepted as on of the most Important goals of the project. During the work one has realized the complexity of stability. However, we have pre sented a criteria philosophy (see section 6) and put much effort Into starting a discussion around operational procedures. Operational  manuals and "practical operational stability" are vital key words.

 program type of mathematical model: How the model does Itteratlons for varying draughts and how 1t finds the vertical stable position and draught (volume and displacement). How a computer program adjusts for longitudinal position of  buoyancy and   corrects for trim  In heeled conditions is also an important factor.

3. THEORETICAL MODELS AND MODEL TEST HESIILIS EROM • FOLLOWING WAVE SITUATIONS ^ 3.1 Stability for vessels balanced  on a wavecrest In following waves. It Is an International understanding that future criteria should reflect physical situations and   the "following waves" Is one to be considered. This situation defines three wave/response pheno  mena which could   be dangerous:

The accuracy and the routines when read ing hull data Is also an important fac tor. As ah  example one  program   was  parallel tested  in UK and   Norway for the same basic hull form.  These results gave different answers.

Loss of stability when balanced  oh a wave crest. Parametric resonance rolling. "Broaching to". "Loss of stability" were likely to be the first physical situation taken Into consideration when proposing criteria. The Influence of waves on transverse stability has been studied  for years In the literature. It 1s well known that the righting levers (GZ) In general are reduced compared   to the still water values when the vessel Is balanced on the wave crest at position near amidship. The righting levers increase above the still water  on a trough. values when the vessel is balanced 

5 DISPLACEMENT 724 m 

S

no relative speed between the wave and vessel Is considered no ship responses are taken' Into account Interaction between the ship wave arid  the actual wave Is not considered OurIng the project work a reliability study was made. In this study we  tested computer pro grams often used  In Norway -  some results are shown In fig. 1. Conclusions made:  Most computer programs produce Identical still water hydrostatics and    righting  moment curves (KY).  When calculating hydrostatic and   righting lever for a vessel balanced  on a wave crest In following seas, differences occur from  one program  to another. This Is mainly caused  by deviation 1n each

; -  147

"X^fe

\ : ~ \   _ • * : . T.rf— N "> "-J '"«^. — \ ^ ^ :  -r'..*••• 

,> •

r^s

   __ 

 _  When assessing criteria for such a   situation  by hydrostatic calculations, this means that

SINUS HAVE HAVE HEIGHT Sm  WAVE LENGTH 43 m  (.|.BC) HAVE CREST AT LBP/2

r

•«

(r

M

*FREOLV".-i>iinr.« ARCHIMr.OES .  PIÏLIK0N

> «.-——*—  

SIK08

Fig. 1.  GZ-curves for a basic model. 3.2 Model tests in following waves.  A model of a typical Norwegian coaster was chosen as the "basic ship". More than hundred vessels of similar size and    arrangement  are working In the  North Sea and Norwegian waters.

 MAIN DIMENSIONS

V4JS"» U5_r-sm:!v :•••-••.?•.: SjSW".:'H:i: -r?  r-..: i--:.zïttWjÊr.

Fig.  2.

Lines - basic hul l form.

The model test was performed  In the towing tank at MARINTEK. The experiment was arranged  partly as a captive test and tests with the model free to roll, pitch and   sway. The model was attached  to a  wagon and run from the end  of the tank In direction following the waves. For each run the constant heeling angle was adjusted' In steps of 3-5 degrees. Starting from upright posi tion, tests were performed for heeling angles up to 600. Only regular waves were tested. They were tra ditionally referred  to as regular sinusoidal. The correct length was achieved  by tuning the period of the wave maker. Some results are shown In fig. 3.

?, I"! 0.5

oiscmcEHEnr m  »• on o.is » — u n lnljht 3 * " * • «• • • Sa

these effects occure on a hull form with flat aft end and great flare In the foreship. A hullform like this Is Initially  more exposed  to hydrostatic loss of sta  bility. For others the "loss of stability situation" must be reconsidered   as not  being dangerous. Ill) Before assessing criteria for "loss of stability" In following waves, the above Important facts should  be considered and studied closer.

4rzr-^
=aS

Hona sPKOiO.a  o/i ^•- 

cj)mi

E

" - -  HAVE CK5T »T W/FP

4 ^ ^ ^ ^

0.4

actual »ave


Fig. 4. Wave Interaction.

0

40° SO" AngU of hn l  <*fl.)

3.3 Broaching to-sltuatlon - theoretical models

Flg. 3. GZ at low forward speed;

In following .waves we  often refer to a dange rous situation named  as "broaching" or "broaching to". The situation 1s described as course Instabi lity. The vessel turns around  In following waves and capsizes as a total transient event. To study this stability problem  we have been working clo sely on the course Instability problems of "broaching to", both through theoretical studies and model tests. Due to the combination  of an external moment from the wave and a reduced rudder effect, an oscillating yaw-motion can occure. Increase uncontrolled and turn the vessel around.

 Non-captive tests were performed with the  mode! running In upright position. All motions In six degrees of freedom were measured. The Inten tion of the tests was to  find  If a  negtlve righting moment would lead  to a capsizing. The stability was decreased  to critical, according to  previous computer calculations and the tests with the captive model. The following main conclusions were made after analysing the results: 1)

The Interaction effects between the seawave and the actual diffraction waves are very  Important factors. These are uniInear and not well known and should  be studied further 1n the future.

In the literature one can refer to several theoretical works trying to explain this situation. From these works we can conclude the \ following:

11) As the Interaction effects dominate the resulting geometry situation, It Is likely that certain types of vessels are  more exposed  to loss of stability In following sea than others, especially If

- 148

1.

The probability of broaching Is greater when the ship Is running down Into a wave trough.

2.

The probability of broaching Increases with Increasing waves.

3.

The resul t of some ca lc ul at ion s' tha t r efe r i n i t i a l conditions of the model test described the following, is shown In fig. 5. They refer calcu latio ns in the yaw-modus and Is taken from

The wavelength must be greate r than the ship length when broaching is l ikely to occure.

In pr in ci pl e, due to great motions and waves, this situation Is complex to understand and analyse. We decided to attack the problem through theoretical analysis of ship manoeuvring In foll owing waves. As we found it Important to eva luate the transverse forces and yaw-moments on the h u l l ,  theoretical models were developed. The first was a generalized strip theory and the other a generalized bow-flow potential theory. The last one was developed for our situation In a computer program and used to calculate the forces and moments of a simp lifi ed hull ( f l a t pl at e) . [2]

N«/ n »

iinn  p w.'

to in to [2 ] .

a

 z?—**"

:~

 .h M 

A

i

Fig.  5. hydrodynamlc coefficients numerically   cal culated for a flat plate In yaw-modi [2]

In the actual "broaching" situation a vessel is running In the waves with speed close to the phase-velocity of thé wave. The wave encounter frequency Is low and the vessel is stabilized in the vertical plane. Making the assumption that non-viscous forces are dominating such a situation, the problem can be described by pote nti al theor y: * (x.. t ) - - UX -fr 4 U . t) The potent ial * (x_, t) describes the v elocity f i e l d around the vessel and have to f u l f i l cer tai n boundary conditions that reflect the free surface and the hull surface.

Fig.  6. Definitions.

The results refer to the plate floating stable in the wave. I.e. stable In vertical direction in different positions In the wave (see table below).

- UX : represents the free flow potential  )  : represents the par t that is caused by the hull Itself and can be derived Into >   - 4>w + +s 4M

: represents Incoming waves

4. s

: represents the vessel s  effect on the waves due to radiation and diffraction

e Xc 0 0 A/4 -0.079 rad A/2 0 3/A/4 0.079 rad

As the actual motions refer to low frequencies only, the stationary generalized bow-flow problem Is considered. The numerical method is then based upon a system of describing the pot ential by a dlpole-distrlbutlon that copes with the given boundary conditions. The transverse force and the yaw-moment are presented 1n the form: Y • YVV * YVV

transverse force

N • NVV • NyV

Yaw moment

3.4 Broaching to-situatlon, model tests Model tests were being performed to verify computed values for the transverse horizontal force- and yaw-moment on a capti ve f l a t plate in varying yaw modi and following waves. Tests were also performed for an actual hul l form with a com parable l at er al area and aspect ra ti o as the plate.

there V and v represents the crossflow and acceleration. - 149 -

The model tests were performed  in the towing tank at MAR1NTEK. The experiments were arranged as a  partly captive test. The models were made free to heave and pitch. They were kept  In the rol 1, sway and surge mode and   attached   to the tankwagon. Through a dynamometer the horizontal force In surge and sway and the yaw moment were  measured (fig. 7). .

' ^ ^ ^ g ^ ^ .-• i1 i>• i1

C:,   •* _

~~—-^

L

3 0

.. =^

////// it  H  11 /

^

>

W**t transducer

Fig. 7. Model test set up The tests were made for varying forward speed. The yaw-angle was varied from  0 to 15° In steps of abt. 3°.

Fig. 8. Definitions of the position  In the waves.

The wave trains were produced as regular waves for a given wave length. The wave length was espe cially chosen to fit the ratio  A/L»1.0 and  A./L=l.45.  The height was chosen to be well below the critical steepness ratio, thus giving rather regular sinusoidal waves. The correct length of the wave was achieved  by tuning the period  of the wave maker.

s-:

Theory

U  N

The tests in the wave train were performed by running the model at varying speed. The measure  ments were recorded  In a time section of constant forward speed after the acceleration period. Referring to the theoretical calculations  the  positions at the plate and the  hull model were defined as shown In fig. 8.  As a general conclusion, the model tests show that the theoretical models calculate forces and  moments In correct range. There are, however, cer tain deviations in the values compared, that need closer studies before the theoretical model can be fully accepted. These deviations could probably be caused  by inaccurate measurement and.defining, of the position In the waves.

• Tests

-

Theory Tests

Fig. 9. Hydrodynamlc coefficients as a function of  position in waves (flat plate. The measured values for transverse horizontal force, the yaw moment and also the longitudinal force show that they are varying with the objects (plate, vessel) position  in the waves.

.'- 150 -:

4. SHIP MOTION AND SHIFTING OF CARGO-

 As seen from fig. 9, maximum yaw-moment occurs when the vessel Is positioned In a trough In abt. 3° yaw angle. At an Increased angle (6°) the situation has changed. The maximum transverse force has Its maximum on the top of the wave crest, and the surfing force from the waves Is also most pronounced at rrest situations.

 An area that requires both stability rules and criteria reflecting physical situations Is trans  portation of bulk cargoes onboard ships. The research programme performed have been looking Into cohesive bulk cargoes in particular and the effect of ship motions upon cargoes in general. The before mentioned-basic ship Is also tested in longcrested and shortcrested seas with dif ferent headings, speeds and stability. The motions  pitch, heave and roll were measured, so also the accelerations In x-y-z directions at different locations In the cargo hold area.  A strip theory computer programme was used for calculating the same parameters in longcrested seaways and with similar wave height as in the  model test. A comparison between the model test results and the computer calculations show a fairly good agreement. The conclusion Is that for the actual (conventional) hull form the computer  programme strip theory 1s well suited for a  motion-acceleration calculation. Another Important conclusion from the tests/calculations is the range of the accelerations In the cargo area. For relatively normal seastates the accelerations, especially 1n the transverse and vertical direc tions, are considerable.

Fig. 10. Hydrodynamlc forces acting on a captive hull model in following waves. Based upon these measurements.and also obser vations made from earlier model tests In similar following wave situations, we can describe the  broaching event as follows: 1.

When a vessel 1s balanced on a wave crest it looses upright stability and Is pushed forward by the wave surfing forces.

2.

Transverse disturbances easily occur as the control-force from the rudder decreases. At small yaw angles great transverse force's makes the vessel change direction.

3.

Running down the wave the yaw moment Increases and the vessel turns around easily, ends up In 90° against the waves.

4.

As this transient event happens fast, the forward kinematic energy transfers to an Inclining moment.

5.

The vessel rolls heavily or capsizes.

 New methods have been tested out for the determination of the slope stability of cohesive  bulk cargoes. The methods are commonly used by soil engineers when determining the safety factors against rupture of different materials. For the cohesive cargoes, which are very complex to handle In the existing rules and recommendations, both cohesion and frlctlonal angle have to be known together with moisture content and porosity of the cargo pile as loaded Into the ship's hold. A testing method Is used (trlaxlal apparatus) to find the cohesion and friction angle, when parame ters like porosity and moisture content are varied. The testing apparatus does also give possibilities of investigating the change In cohesion and fric tion over time when the sample (I.e. the cargo  pile) 1s exposed to several cyclic loads. Tests show that the cohesion will vary quite a lot over time for some materials. The test-results from the trlaxlal tests have  been used In connection with the motions and acce lerations of the basic ship with a certain load configuration, stability and wave condition.

- 151 -

Applying D'Alembert's principle, the acceleration forces exerted to a particle within the cargo pi 1* can be expressed expressed as D'Alembert's D'Alemb ert's forc fo rce: e:

By making use of the motion and accelerations 1n fig. U, the safety factor against sliding failure Is calculated for a full scale cargo  con figuration of the "basic" vessel. Copper  con centrate with high and low porosity is calculated. The f r i c t i o n angle and and cohesion cohesion are decided out from from tr ia xl al test te st s, and and the resu lt from from the the calc ula tio n Is pres presen ented ted In f i g . 12. 12. A ruptu re, or sliding failure will take place when the safety, facto fa cto r Is lower lower than 1.0. 1.0. I t w i l l be seen from from the curves curves that th at the contri but ion s from r o l l and acce acce lerations are presented separately, and that the change In safety factor Is rather high when one of the parameters are Included/excluded.

"A mass pa rt ic le (ro), w i l l be In eq uilibr uil ibr ium when exposed to an acceleration (a*g) by a force (f) equal to (m*a*g) in the opposite direction". Applying this principle, the methodology of a dynamic problem Is reduced to a static one. In this way, the ship Is turned a certain angle to a st at ic angle angle whic whichh corr corresp espond ondss to the to ta l force from accelerations In transverse and vertical direction and from roll motion. In this context, 1t should be mentioned that the pitch motion and the longitudinal acceleration 1s not taken into account. The reason for this 1s that these parame ters are small compared to the others. Using D'Alembert's principle, the slope stability of a cargo pile 1n "rest" can be judged when set into motion during a sea voyage. The safety factor can be calcula calc ulated ted fo r a given sea sea state and a given heading for the ship when the necessary motion parameters are known. F 1 g . l i shows the roll motion,  transverse and and vert ve rt ic al accelerati accel eration on as as a function of phase angle for a certain wave  con dition of the "basic" ship. The motion and acce lerations are given as significant values for a significant wave height of 5.0 m, heading 90 degrees. The parameters represent a time Interval of 30 minutes in the actual seastate.

n - rotation of cargo cargo pile from roll motion motion 8 • rotation of cargo plia from acceleration»

Safety against rupture

* •

inct. roll and ace. Inc., roll

9

incl. ace.

a-*

it •

•-

/

•au. Itwu]

. M

• •

/

•

•

/i

M

/l»

> Rupture

\

/

1» ftUU ABOLI (0»|.)

\

0

to •

 \\  \

ti

It t ph

/ tBAMIWtMt «C

I

ts -

""

/

/ \ _ j —

X |

Starboard side of carg cargo o pile pile

*  incl. roll an  roll and d ace. o  incl.  roll a  Incl. occ.

1-

•i

™S>

Varatian of rupture safety as a function of phase angle,high porosity

•sro

• >h

"»°"fl'-  '*> j.

lo 

 \

**

Me

1'

I • 1 •

< • »

"ft •

I

m

i«b

I' ram ram in g u, t ,y.

/

Phase Phase angle

1.

Varaticn of rupture safety as a function of phase of  phase anglejow  anglejow porosity

•'

• % Bttsutg Bttsutg » &w «M CtttB CtttB 0 luwti

Fig.  11. Significant roll.motions and acceleralons as function of phase angles.

- 152 -

Fig.  12. Safety factors against sliding failure for copper concentrate.

Comparing the upper and lower diagram, one Will  Will  observe the change in safety factor with the variation of porosity. With high porosity In the cargo pile, a sliding failure would take place during the 30 minutes voyage with bean seas, while the cargo cargo wit h low low porosity poro sity woul wouldd not sh i f t during the sane time Interval. A computer program Is developed to give the master  master  of the ship the necessary knowledge about the cargo cargo an and the safety with respect to sh if ti ng In varying seastates. 13 show showss an an example example o f some o f the outp ou tput ut from this programme. The ship master will have Information e.g. about the variation In safety factors for different loading configurations. For the given exam exampl ple one one w i l l notice not ice tha t hatt the safety saf ety factor will  will  be less than 1.0 for cargo pile heig he ight ht above above 6.0 m. The The computer computer prog progra ramm mmee also al so gives possibilities to judge variation In material parameters, parameters, seastates, cargo cargo configurati configu rations ons et c. Fig.

 Most cargoe cargoess will will shif shiftt if the-weather con ditions. I.e. ditions. I.e. the motions and accelerations acting upon the cargo are severe enough. However, In most cases the ship's master has poor knowledge of the limits when cargo starts shifting and the forces which Is acting on the cargo.  New rules rules and and criter criteria ia shou should  ld  be  be established In a way that gives the master of the ship optimum Information about ship stability, cargo proper ties and dynamic behaviour of ship and cargo. This Information should lead forward to a guide for the  master on how how to to handle handle the the ship ship In In severe, severe, dange dange rous sea conditions. 5. ENVIRONMENTAL  - A   BRIEF PRESENTATION OF NORWE BRIEF GIAN WAVE RESEARCH WORK The Initial plans of the program  on wave research have not been realized. Due to changes In the financing situation and the fact that parallel research work has actually been going on sponsored  by the the oil-ind oil-industry ustry,, these these plans plans were deleted deleted.. Under this program umbrella we have kept a close watch on this research. res earch. Results have been  presen  presented  ted  In the steering committee and   given  given and   moral  moral support support when when needed. needed. At MARINTEK breaking waves have been studied  for several several years yea rs..  The statistics of breaking waves have been studied studied through re-analysis of wave signals from wave rider buoys positioned along the Norwegian coast. .

Fig. 13. Factor of sliding safety with respect to height of cargo pile.

the  following

can be

Cargo shifting onboard vessels Is causing a lot of accidents and the probability  of such occuren ces should  be better documented  via methods as described described or similar methods.

[

*

]

•

In the laboratory new testing techniques  for generating transient waves have been developed. The last generation of these Is a  non-linear experimental technique for deterministic generation of freak waves both In 2-d1mens1onal 2-d1mens1onal (longcrested freak waves) and in a directional 3-dimenslonal short-crested wave field. This technique has also lead  to  to a better understanding  of the physics of non-linear gravity waves. It explains why the con ventional statistics used  to  to derive a "100-year" design wave from  a limited amount of data obtained with waverider buoys,  do contain  do not necessarily contain the event of the extreme freak wave.   [5]

elopi iwight •

 As a  conclusion, recognized:

•

 When  When perform performing ing survi surviva vall testi testing ng of of vessel vesselss  in  in waves to measure stability, this th is wave generating technique has lead from stochastic testing Into deterministic testing of vessels using a design  philosop  philosophy hy as show shownn on the the fig. 14 14 below below..

- 153  -

Extreme freak wave breaking

ExtremeIresponse

[

 We know know that that for for a vessel, an Important part of the operational advice Is given In the stabi lity booklet, where advice Is given on how to obtain a specified stability (expressed  by Gx and Gm) under all loading conditions considered rele vant.

Probability for selected wave

Critical event ?

J

J

However, this stability Information Is not related to environm environmental ental conditions. conditions . Furthermore, no advice Is presently given as to how the probabili ty of capsizing capsizin g can be kept at a relatively low level.

 for failure Probability for Fig. 14. 14 . Design philosophy. phi losophy. On the subject of foreca forecasti sting ng dangerous dangero us wave situations, work has been performed by MARINTEK In cooperation with the Norwegian Meteorological Institute. A  practical operational forecasting forecasti ng  model  model of dangerous wave situations along the Nor wegian coast has been developed. 24 different areas along the coast were found  found  to be  more exposed than others   (the Ship in Rough Seas Project).  Examples from two  of these areas have  been chosen chosen to be observed  In detail In order to [6] evaluate the forecasting model.  

 We Inte Intend nd oper operat ation ional al proced procedure uress or manua manuals ls to  be  be the tool of the  future In answering these questions and the one  to combine both the human, h uman, environmental environmenta l and design factors In In a total stabi lity and safety concept. Examples of operational operational manuals have been developed during the the program  and discussed with  people  people concer concerned ned [7]. [7 ].  Our latest version of a  manual  manual Is built up of three chapters which  are  In Individual  presen  presented  ted  In Individual booklets book lets::

In one of the areas a capsizing of a coaster and the following Inquiry lead  to a closer study of that particular area. The meteorologists have, however, not yet finalized the evaluation of their  model. When When the the time comes, a radio forecast will  be devel develop oped ed especi especiall allyy suit suited ed for for the the coast coaster er and the fishing vessel traffic.

One brief emergency part of 2-3 pages One operational part par t that describes describe s the  most Impo Import rtan antt Items Items In  operational stability. These are: • a weathertlght vess vessel el •  stability and loading conditions practical control of the actual actual stabi •  practical lity during a journey • how to judge a critical situation In waves •  safety actions to control and Improve the tightness   stability and  • damage stability • using working tools onboard (fishing gear I.e.) • cargo handling •  evacuation plans

6. OPERATIONAL OPERATIONAL PROCEDURES AND MANUALS The basic stability philosophy behind  the  progra  programm idea idea mention mentioned ed earlier, earlier, consists consists  of three  main  main parts. parts. One One of them them Is oper operati ationa onall procedure procedures. s.  We feel feel that that this this theme, operational stability, should  be studied closer In the  future  and be Integrated Into criteria definitions. We  have often asked If the general safety of a vessel will  be Increa Increased sed If only only new new design design require requiremen ments ts simi lar to the ones  In sophist i  In force today but more sophisti cated, are developed. Is It  possible to plan and design vessels that never capsize without also setting operational criteria? Or Is It possible through bad semanshlp to capsize any vessel?.

- 154  -

One general general part that presents general general Information to the master, such as: • the basic principles of stability • the vessel vessel motion characteristics character istics In waves '•• • cargo handling • wave and weather Information. Information.

the manual should contain In principle the c ontain  the state of the art of stability knowledge kn owledge expressed In a simple manner. m anner. In this way It also has an educational aspect that over time will improve safety.

LITERATURE 1.

T. Nedrelld: "S tab ili ty f o r Vess Vessel el s  s Balanced  Balanced on a Wavecrest 1n Following Waves, Hydrostatlcal Calculations - Model Tests." NHL-report [1985].

2.

T. Utnes, Utne s, NHL/ NHL/VH VHL: L: "Calc "C alcul ulati ati ons of Hydrodyn Hydrodynaanlc coefficients for Vessels In Following Waves (In Norwegian) [1985].

6.

P.  P.  Schel berg. Schel berg.   Meteorological Institute, S.P. Kjeldsen, MARINTEK: "Belgevarsllng - program for ver lf1 sering. Meteo Meteorol rolOgl Oglsk skee In st lt ut t/ MARINTEK [1984].

7.

Emll Aall Dahle, T. Nedrelld: "Operational Manuals for Improved Safety in a Seaway". Stab. [1986] Polen.

3.

4.

5.

T. Medre lld, O.E. Rel tan:  "Measuring of Hydrodynaralcal Coefficients for Vessels In Following Waves; Waves; Model Model t e s t s . " MARINTEK [Dec. [D ec. 19 85 ].

8.

S.P. KJeldsen, 0. Myrha Myrhaug: ug: "Parametric Model Model ling of Joint Probability Density, Distribu ti on s fo r Steepness Steepness and Asy Asymm mmet etry ry 1n Deep Deep Water Waves." Applied Ocean Research [1984] Vol.  6, No. 4.

F. Fred rlksen , E. E. Jullumst re: "Sli ding Failures of Bulk Cargoes During Sea Transportation". MARINTEK report no. 520021 [1985].

9.

F. Fredrlksen, A. Kavlle, E. Jullumstre: "BUL "BULKC KCAL ALCC-A A Comp Compute uterr Prog Progra ramm mmee f o r Calc Ca lcul ulat at ion io n of Safety Factors Against Rupture of Cohesive Bulk Cargoes". MARINTEK report no. 530094 [1986].

S.P. S.P . KJeldsen: KJeldse n: "Dange "Dangero rous us Wave Groups". Ma ri ri  time Research No. 2 [1984] Vol. 12.

10.   E. Jullumstre: "Shifting of Cargo and Stabi lity Criteria of Small Cargo Vessels". NHlreport no. 183225 [1983].

- 155 -

Third International International Conference on Stability of Ships and Ocean Vehicles,  Gdarisk, Sept.  1986

0

"o,

o)

(0)VO (0)VO

Paper 4.1

PROBABILITY OF CAPSIZING IN STEEP WAVES FROM THE SIDE IN DEEP WATER by E. Aa. Dahle *) and D. Myrhaug **)

ABSTRACT A model for estimating the probability of capsize In steep and high waves from the side in deep water is presented. The main element of the model is the estimation of probability of occurrences of steep and high waves for given sea states by using the joint probability densitydistribution of crest front steepness and wave height. The sea sta tes are described by usin usingg a joint frequency frequency distribution of significant wave height and mean zerocrossing period. The dominant wave direction is taken from wind statistics. Then, the operations of the vessel have to be investigated in order to assess when it is exposed to the waves, and and its loading loading conditions conditions and associated stability. stabili ty. The stability has then to be associated with a critical wave height, using model experiment data. Finally, the human element with regard to maintaining the stability (i.e. closing openings and securing cargo) should be assessed. An example from Norwegian waters has been chosen in order to illustrate the application of the model.

2) shelter shel ter areas a reas shall be intact int act during a DAE DAE until safe evacuation is possible. 3) depending on the platform type, type , function function and location, when exposed to the DAE, the main support structure must maintain its load carrying capacity for a specified time. In this paper this approach will be compared with the capsizing accident. MODEL L 2. THE PROBABILISTIC MODE First, the definition of "capsize" used in this paper is illustrated in Fig. 1. Clearly, only the "stable side position" may comply with the 3 criteria listed above.

a. Self-righting. Capsize not feasible

1. INTRODUCTION In 1981, The Norwegian Petroleum Directorate issued; its first Guidelines for safety evaluation of platform conceptual design [l]. In these guidelines, those responsible for the design "shall verify a sufficiently sufficien tly low probability of loss of of human life, high material damage and unacceptable environmental pollution as a consequence of the accident". The types of accidents should be relevant to the design under consideration and should have a probability of occurrence above 10"* per year if their consequence is to be considered. In such cases, the accident is denoted a "Design Accidental Event" DAE. The design must then be such that only personnel in the immediate vicinity of the accident are endangered. This requirement is considered as satisfied provided that the following 3 criteria are complied with; 1) at least lea st one escape escap e way way from from centra cen trall positions which may be subjected to an accident, shall normally be intact for at least one hour during a DAE.

STABLE

|.

STABL STABLE E

J

b. Capsize to 180° 180°

^STABLE JSTABLE^ STABLE

c. Capsize to stable side sid e position position Fig. 1. Definition of capsize.

157

A method f o r  estimating probability of   capsize \yas

Table i .

first presented by  Sevastianbv [ 2 ] .

Joint distribut ion of H s and T-, (based o n data from Krogstad  [5]).

This paper follows th e principles  of [2] in estimation of

n>

the probability o f  capsize in   steep, near-breaking waves f r o m t h e side i n  deep waters. This event is  considered  to be

1-5

5-6

6-7

7-Ï

8-9

0.096 )

0.1315

0.07 38

0.0277

I   

0.0015

0.0508

0.0612

0.0017

9-10

10-11

Hs(ml

; relevant t o  smaller ships,  i.e.  with length below   45 m. As  for the estimation  of the  probability of   this event,

1.5-2.5

0.005«

0.0015

0

0.0311  ; 

0.0022

0.0003

0:0325

0.027)

0.0085

0.0015

0.0001

0.001(7

0.0^06

0.0078

0.0013

0.0001

5.5-6.5

0.0066

0.0092

0.0015

0.0002

6.5-7.5

0.0005

0.0039

0.0019

0.0001

0.0010

0.0018

0.0001

0.0007

0.0002

only an  order of   magnitude  can be arrived a t .  Obviously, this 7..5.J.5

is also t h e case f o r  accidents in the oil   industry. The aim of this paper  is to compare t h e probability o f  capsize with  the

3.5  il.5

. target; value   of 10- * per  year.

1.5-5.5

f  

In th e paper, th e  probabilities  of the   different elements o f capsize i n deep waters a re dealt wi th in

succession.   First, t h e joint probability of   encountering a steep a nd high w ave in a given sea  state is  covered, and the

7.5-8.5

probability of   capsize after such an  encounter is  discussed;

-

Then,  operational aspects a r e brie fly covered. Real istic

•

8.5-9.5

estimates o f  probabilities of   adherence t o  operational 0.0004

>  9.5

requirements,  based o n knowledge  of the ship  and its  crew, should be  include d.

Then th e level  of "bad weather" must be  defined. A '

Finally, th e  total probability of   capsize in   deep waters

natural choice will  be the  significant wave height H . The

Is calc ulate d.

choice depends o n  vessel size.   H > 1.5 m is chosen i n  this paper.

3.   PROBABILITY O F EXPOSURE  T O B A D WEATHER

The joint probability distribution o f  significant wave By investigating th e operations o f a  ship, th e  period of

height H s a nd   mean zero-crossing period T z  for H s  > 1.5  111

time f o r  which  it is exposed can be determined.  For a

is given in  Table 1. This table is  obtained from th e  joint

fishing vessel, f o r  instance, three operational phases   are of

frequency table of H s and T z  given i n Krogstad  [ 5 ] .  This

interest,  i.e.

data is  obtained by a  Waverider buoy covering t h e  period

in transit from th e harbour  to the  fishing ground -

on the   fishing ground

-

in   transit from t h e fishing ground back   to the

1974-80 at   Halten  on the  Norwegian continental shelf. Finally, t h e duration o f  various sea  states is  given in Kjeldsen [é] by a  Weibuil distribution. From this,   the

harbour.

average duration of   storms f o r  winter conditions is   obtained

Only a  closer study ca n reveal t h e  phases o f

as  ,(see  Table 2 )

interest as  indicated b y Dahle an d  Nedrelid [ 3 ] .

"C(H S )  = 1.27- t r

The direction  o f t h e wind  in the  area under

(1)

S t r i c t l y , Ï is a function o f  both H s an d T 2 , but to the

consideration  may be  important and may  o f t e n be  available

authors knowledge this information is not yet  available.

as a  direction frequency diagram. Assuming near

From this data, a weight function  w is set to 1.0 if

correspondence between wind a nd wave direction,k nowing the course  of the vessel, a n d defining an  exposed sector o f

the duration  o f t h e exposed peri od (t ) is exceeding the

90 degrees width on  each side  o f t h e ship, a probability o f

duration  of the sea state. Otherwi se, w is  taken as the   ratio

exposure can be arrived  a t .

t/Tfj.

Such "wind roses" f o r  various areas  may be given on a .yearly or   monthly bases. F o r  Norwegian waters, t h e  former

Table  2. t ç and X as functions o f H s > fr om Kjeldsen [ é j .

data a r e given by   Haaland [4J. An example is  shown in  F i g . 2. The probability o f weather from t h e side, P i , is  then t h e  integrated wind

Hs

frequency distribution  for 90  degrees, on  each side  of the

(in)

"E

*c

(hours)

(hours)

ship.

Wind rose Area = A

shaded area

Fig. 2.   "Wind rose" and P)

1.5-2.5

35

44.5

2.5-3.5

17

21.6

3.5-4.5

9.6

12.2

4.5-5.5

6.3

8.0

5.5-6.5

4.4

5.6

6.5-7.5

3.3

4.2

7.5-8.5

2.5

3.2

8.5^9.5

2.0

2.5

>  9.5

1.7

2.2 .

158

Then the general expression for the probability of being in a situation Sj during a one-year operation of the ship is:

ASYMMETRIC WAVE OF FINITE. HEIGHT

P(Si)=SSlPljk-P2jk- ^jk*P3jk- P*»jk <2)  j k where j and k denote summing over significant wave height and zero-crossing period, respectively. Further p ljk = s e e F>S- 2P2jk = yearly fraction of time of exposure in sea state  jk. = P3jk  P ^c» ^cI  Hsj> TzU)' l 'e'tne  conditional probability of steep (8> 6 C) and high (H > Hc) waves for a given sea state (H  sj, T2|<), see section *. = o mt Pfjk J   probability of H s, T z, see section *.

Fig. 3.

The probability of capsize caused by a steep near-breaking . wave in deep water is then P c = P(Sj) • P 5 • Pé • F7 <3> where: P5 = probability of being hit by a steep near-breaking wave during the period of rolling when the vessel is most exposed. In this analysis, P5  =   0.5. Pg = conditional probability of capsize, given an extreme wave situation. According to the discussion in section 5, Pg is 0 for the "safe" regions of Fig. 5, and i.O for "unsafe" regions. • P7 = conditional-probability of compliance with the requirements that makes a ship "safe", i.e. the probability of important openings being closed properly, and cargo secured to sustain large heeling angles in the situations under study.

where \' is the crest elevation measured from thé mean water level, T' the time defining the position of the Wave crest relative to the zero-upcrossing point in the time domain, T the zero-downcross period and g the acceleration of gravity. The definitions in the time domain, also of X and it, are shown in Fig. 3. It is generally accepted that use of the crest elevation for design applications provides a basic j parameter more relevant to finite amplitude wave geometry' than the wave height. Observations of breaking waves show that these waves can be characterized by a very steep crest front and high asymmetry factors. The  E -parameter is thus a mean crest front inclination in the time domain. However, it is not sufficient to describe the wave / conditions by steepness and asymmetry parameters alone, but they should be combined with the wave height ,to give a much better description of the severeness of a given sea state. Joint probability density distributions for £ , H and X, H are given in Myrhaug and Kjeldsen [8]. Myrhaug and Kjeldsen fö) discuss closer the £-H distribution, see also Myrhaug and Kjeldsen [l0]. The joint probability density destribution of crest front steepness and wave height, p(£, n), is obtained as a best fit to field data records from the Norwegian continental  shelf.  Here 6 = £/&rms and h  = H/H rm s are the normalized crest front steepness and wave height, respectively. 6 r m 3 and H rm s are the rms-values used for normalisation. The joint probability density distribution is determined by p(6,h) = p(ê|h)p(fi) A  (5) Here p(h) denotes the marginal distribution of h and .p(ê|h)L;denotesthe conditional distribution of £ given h. The joint probability distribution is fitted to the actual data by first fitting the conditional distribution of £ given h and then fitting the marginal distribution of h. The data represent 58 time series, each of 20 minutes duration, from measurements at sea on the Norwegian continental shelf including altogether 6353 individual zerodowncross waves. These data were taken from a larger data base, see Myrhaug and Kjeldsen [8], sampled in the period 1974-78 with three Waverider buoys located at Tromsflf laket, Halten and Utsira. The properties of the obtained probability density distributions show that data

*• PREDICTION OF OCCURRENCES OF STEEP AND HIGH WAVES IN DEEP WATERS  HA 3oint distribution of crest front steepness and wave height A method for estimating encounter probabilities of occurrence of steep and high waves in deep water for given sea states will be described briefly in this section. The method is based on the idea given in Kjeldsen and Myrhaug [7] utilizing the advantages that are contained in a zerodowncross analysis by using the wave trough and the following wave crest in the definition of a single wave, and defining the wave height as the difference between thèse water levels, Fig. 3. The zero-downcross analysis is the only analysis which provides parameters that give a representation of the physical conditions with relevance to breaking waves, and thus, the only parameters which should be correlated with measurements of severe ship responses or wave forces in such waves. Further, a more accurate description of steepness and asymmetry in transient nearbreaking waves were obtained by Kjeldsen and Myrhaug [/J, when the three following parameters were introduced: crest front steepness £ , vertical asymmetry factor A and horizontal asymmetry façtorit,. The crest front steepness is defined by , e =

 (g/2Tr nv

Basic definitions for asymmetric waves of finite height (from Kjeldsen and Myrhaug [7]).

w 159

obtained from the three locations can be regarded as belonging to the same statistical population. This means that common-statistical distributions can be obtained, which are representative for the wave dynamics in the whole area (Kjeldsen and Myrhaug OQ). • Two different parametric models were fitted to the data. These two models are based on the fit by a Weibull and a log-normal distribution, respectively, to the conditional A

NORMAUZEO CREST FRONT STEEPNESS S

0

10

n^3^tet

V  s

'^#v

A

distribution histogram of £ given h. In Myrhaug and Kjeldsen [9] is shown that the latter model is closer to the trend In the data for higher values of crest front steepness and wave height, and this model will therefore be used in

3r

(a) NORMALIZED CREST FRONT STEEPNESS I 10 20

A

T T

JO

this analysis. The marginal distribution histogram of h was fitted by a Weibull distribution. The Weibull probability density distribution of h is A A  J  given by p ( h ) n3  k^P -li e X p, ^ K].aC , >A 0 (6) with the Weibull-para meters ? =1.05 and p = 2.39 (7) The log-normal probability density of £ given h is given by (b) -

Fig. k.

A

z

where the mean value B and the variance Ï of  In £ are given by • 0.02* - 1.065 h + 0.585 h2 for h < 1.7

(9)

P[(£>£c)n(H>Hc)|£rms,Hrms]

0.32 arctg[3.14 (h-1.7)] - 0.096 for h  > 1.7

0»

and Jf 2(h) = -0. 21 arctg[2.0(fi-l.»)]+ 0.325 (10)  a r| The rms-values used herein for normalisation, £ rms d Hrrns, are related to wave spectral parameters by 6

and

Joint probability density distribution of 6 and ft., (a) observed joint distribution; (b) fitted joint distribution based on log-normal model for p(l|H).!

rms = 0*0202 + 32.4 « ; & =• m2 gl/mb*

(11)

H rrns = i..8582 Vmö (12) respectively, obtained as the best fit to the data by linear regression analysis, mo and m2 are the zeroth and second moment of the one-sided wave energy spectrum S(f), respectively, defined by m n = J f nS(f)df, n = 0,2, where f is o

the frequency, it is related to a steepness parameter based on the significant wave height estimated from the spectrum, Hs « * "Vmô, and the average zero-crossing wave period estimated from the spectrum, T z = V m o/ m 2i iie *
A

the fitted joint probability density distribution of £ and h as given by Equations (5) - (10). <(.2 Estimâtes for probabilities of occurrence of steep and high waves for a given sea state Estimates for probabilities of occurrence of steep waves will now be calculated by using the joint distribution of £ and h. The probability of occurrence of waves with £ > £ c and H > Hc  for given rms-values of £ and H are given • by

Oo

«9

So

= J  Jp(£,H)dHd£ = J  fi /p(ê,R)dhd£ (13) . Since £ r m s  and H rm s are coupled to spectral parameters this is a conditional probability given a sea

!

state. In Myrhaug and Kjeldsen [9J, [10] the sea states are described by a 30NSWAP spectrum. In this analysis sea states with Hs  > 1.5 m will be considered. Further, for a given sea state an "extreme wave" is defined by the following threshold values of crest front steepness and wave height & c = 0.25 and H c = 4 m (1*0 The choice of H is in accordance with the example discussed in section 6. However, in a general case H c has to ' be given in correspondence with the stability characteristics, for instance the energy E as discussed in section 5 and indicated in Fig. 5. The critical value £ c is not too well known, but current research Is aiming at resolving this matter. The asymmetry factors A and ytt are also important for a proper "extreme wave" description. The sea state is described by H s and T z . Thus, the probability given in Equation (13) is a conditional probability given Hs and T z , that is P3 = P[( £ >  0.25) H (H >  « m) | Hs, T z ] (15) P. is given in Table 3 for the sea states in Table 1.

160 -

In Fig. 5, where capsize and some non-capsize cases are plotted, a tentative curve has been drawn. A distinction has been made between vessels with bulwark and vessels with rail, because a marked capsize difference has been observed with low values of stability. The "stability" is expressed by the internal work E ideally done by the vessel until GZ becomes zero, i.e.

Table 3. Conditional probability of "extreme waves" for .given ..sea states.* »J

M

I •

S.I-I0-* »

« '

»

•

JJ

jj

M

«.J

JJ.CO-J


«.|0->

«J.lfr*

«j-in-»

I.MO-J

M-I0-*

WHO-'

IJ-I0-'

M-IO-J

I.9-I0-'

5J-I0-«

I.I-I0-*

JJ-I0-'

 Î. M0 - '

1*10-'

3.H0-»

I.J-I0-»

CSJ'IO-*

1.S-I0-'

14-10-i

j.l-10-'

M-10-'

IJ-I0-Î

XM0-1

IJM0-'

5*io-'

WHO--

2.W0-'

I.I-I0-»

M-10--

1*10-»

».»•I0-1

' •

.( » • » 10

I •

10.)

»J

 f -

È = û S GZ d«f (tons • m • degrees) (16) A = displacement (tons) G? = righting arm (m)  tf v = heeling angle (degrees)
U-I0-'

3.

PROBABILITY OF CAPSIZE WHEN HIT BY A • •• ; • ' " i i .—i « i» m i • • . i i . ii BREAKING WAVE FROM THE SIDE

I this section the probability of capsize when exposed to breaking waves from the side is discussed. The problem is only relevant for smaller vessels, and published work is scarce. Notably, Dahle and Kjaerland [12], Kholodin and Tôvstikh [lj|, Balitskaya [l»], Sevastlanov [2] . and  Hirayama and Yamashita fl  5j have given some data. In Norway the results from extensive model tests with 3 different vessel types were published as part of thé SIS..project In 1983 by Nedrelid et aL [léj. In the tests metasentric height, displacement and extent of super structure were varied in breaking waves of different height H c . Although not measured, it is assumed that £ for the breaking waves in these experiments were above the lower limit of 0.25 used in the present analysis.

such.x»ses.

As expected, models with positive GZ-values extending beyond 90 degrees never capsized in waves of  H = 10 m.

H c (m)

Av< A=/o2-d