Motion Optimization of Semi-Submersibles

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OFPSISORE TECHNOUJGP CONFERENCE 6200 North Central Expressway Dallas, Texas 75206 THIS IS A PREPR7NT --- SUBJECT TO CORRECTION PAPER NUMBER OTC 7627 Motion Optimization of Semi-Submersibles H. L. Minkenberg and M. F. van Slui.js, Netherlands Ship Model Basin O Copyright 1972 Off shore Technology Conference on behalf of the American Institute of Mining, Metallurgical., Petroleum Engineers, Inc., American Association of Petroleum Geologists, American Institute of Chemical Engineers, American Society of Civil Engineers, American Society of Mechanical Engineers, Institute of Electrical and Electronics Engineers, Inc., Marine Technology Society, Society 02' Exploration Geophysicists, and Society of Naval Architects & Marine Engineers. This paper was prepared for presentation at the Fowth Annual Offshore Technology Conference held in Houston, Tex., May 1-3, 1972. Permission to copy i s restricted to an abstract of not more t h m 300 words. Illustrations may not be copied. Such use of an abstract should contain conspicuous acknowledgment of where and by whom the paper i s presented. ABSTRACT Offshore operations have been performed successfully during the last decades, in which time period it was established that semisub- mersibles are superior t o conventional ship-shaped barges in view of minimum down-time requirements. To meet these requirements it is necessary t o reduce the motions, which can be attained by optimization of the dimensions or the shape of the underwater hulls. For optimization purposes there is a need for theoretical methods in order to reduce costly and time-consuming model experiments. Such a method has been developed a t N.S.M.B., which makes it possible to calculate wave forces, moments and the resulting motions of semisub- mersible platforms of arbitrary shape. The paper outlines the fundamentals of the method. A comparison has been made between computed heave and model heave measurements f o r four cross-section types of lower hulls of a simplified platform in beam waves. Both results show a very good agreement. The results indi- cake that heave will be strongly influenced by the cross-section of the lower hulls in particular with respect to wave period. The paper concludes with a motion comparison for an actual platform, which will be optimized for two operational areas where prevailing weather conditions differ substantially. References and illustrations at end of paper. INTRODUCTION Because of an increased demand for energy and minerals, the search for natural resources is i n a phase of great expansion, Up to 193&, oil, gas and minerals were searched for on land or at Mand waters. In the last decades, however, these activities were extended to offshore areas. From t h e start, exploration and production was restricted to rather shallow water in which one could operate from fixed platforms, using conventional shore-based facilities. The performance of fixed platforms is only to a minor degree affected by waves, wind and current; the resulting deflections are too small t o be detrimental. At present, activities are successfully performed further offshore 5x1 deeper water and rougher seas where only float- ing structures can be used. Utilization of conventional equipment has diminished, since it became impractical in v2ew of its limitations in storm seas. The more expensive equipment and the higher costs for transportation of the products, necessitated designing floating units from a point of view of minimum down time. The limiting factors for platform performance are, in general, the motions of the structure induced by combined external forces. A

description

Motion Optimization of Semi-Submersibles

Transcript of Motion Optimization of Semi-Submersibles

Page 1: Motion Optimization of Semi-Submersibles

OFPSISORE TECHNOUJGP CONFERENCE 6200 North Central Expressway Dallas, Texas 75206

THIS IS A PREPR7NT --- SUBJECT TO CORRECTION

PAPER NUMBER OTC 7627

M o t i o n O p t i m i z a t i o n o f Semi-Submersibles

H. L. Minkenberg and M. F. van Slui.js, Netherlands Ship Model Basin

O Copyright 1972

Off shore Technology Conference on behalf of the American I n s t i t u t e of Mining, Metallurgical., Petroleum Engineers, Inc., American Association of Petroleum Geologists, American I n s t i t u t e of Chemical Engineers, American Society of Civ i l Engineers, American Society of Mechanical Engineers, I n s t i t u t e of E l e c t r i c a l and Electronics Engineers, Inc., Marine Technology Society, Society 02' Exploration Geophysicists, and Society of Naval Architects & Marine Engineers.

This paper was prepared fo r presentation a t the Fowth Annual Offshore Technology Conference held in Houston, Tex., May 1-3, 1972. Permission t o copy i s r e s t r i c t e d t o an abs t rac t of not more t h m 300 words. I l l u s t r a t i o n s may not be copied. Such use of an abst ract should contain conspicuous acknowledgment of where and by whom the paper i s presented.

ABSTRACT

Offshore operations have been performed successfully during the l a s t decades, in which time period it was established t h a t semisub- mersibles a re superior t o conventional ship-shaped barges in view of minimum down-time requirements. To meet these requirements it i s necessary t o reduce t h e motions, which can be a t ta ined by optimization of the dimensions o r t h e shape of the underwater hul ls . For optimization purposes the re i s a need f o r theore t i ca l methods in order t o reduce cost ly and time-consuming model experiments. Such a method has been developed a t N.S.M.B., which makes it possible t o calcula te wave forces, moments and the resu l t ing motions of semisub- mersible platforms of a r b i t r a r y shape. The paper out l ines t h e fundamentals of the method.

A comparison has been made between computed heave and model heave measurements f o r four cross-section types of lower h u l l s of a simplified platform in beam waves. Both r e s u l t s show a very good agreement. The r e s u l t s indi- cake t h a t heave w i l l be strongly influenced by t h e cross-section of t h e lower h u l l s i n par t i cu la r with respect t o wave period. The paper concludes with a motion comparison f o r an ac tua l platform, which will be optimized f o r two operational areas where prevail ing weather conditions d i f f e r substantial ly.

References and i l l u s t r a t i o n s a t end of paper.

INTRODUCTION

Because of an increased demand f o r energy and minerals, the search f o r na tu ra l resources i s i n a phase of great expansion, Up t o 193&, o i l , gas and minerals were searched f o r on land o r a t M a n d waters.

I n the l a s t decades, however, these a c t i v i t i e s were extended t o offshore areas. From t h e start, exploration and production was r e s t r i c t e d t o ra the r shallow water in which one could operate from f ixed platforms, using conventional shore-based f a c i l i t i e s . The performance of f ixed platforms i s only t o a minor degree affected by waves, wind and current; the resu l t ing def lect ions a r e too small t o be detrimental. A t present, a c t i v i t i e s a re successfully performed fu r the r offshore 5x1 deeper water and rougher seas where only f loat - ing s t ruc tu res can be used.

Ut i l i za t ion of conventional equipment has diminished, since it became impractical in v2ew of i t s l imi ta t ions i n storm seas. The more expensive equipment and t h e higher cos t s f o r t ranspor ta t ion of t h e products, necessitated designing f loa t ing units from a point of view of m i n i m u m down time. The l imi t ing f a c t o r s f o r platform performance are, i n general, the motions of t h e s t ructure induced by combined external forces.

A

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11-132 MOTION OPTIMIZATTON OF f3EKtSUBMERSIBLES OTC 1627

I l m e forces acting upon f loa t ing s t ructures operational areas, where environmental c o d -

can be s p l i t i n t o a s t a t i c part caused by t ions d i f f e r appreciably t o detect mutual merits. 1 steady wind, current and wave-drif t i n g forces, and a dynamic par t resul t ing from wave action. FUNDAMENTALS OF THE m O R Y

held within permissible W t s by an appro- p r i a t e se lect ion of the platform submerged hulls .

The consequencss of the s t a t i c forces can be counteracted by a well designed multipoint mooring o r a dynamic positioning system, Oscil- l a t o r y motions, which a re induced by the dynamic par t of t h e wave forces, can o d y be

Since minimum downtime i s of prime impor- tance, a platform should be designed wLth long na tu ra l motion periods t o avoid o r minimize resonance phenomena. Long natural motion periods, outside t h e range of the dominant wave periods, can so le ly be real ized by f l o a t e r s having a s m d . 1 r a t i o between waterplane area and displacement, Column-stabilized semi- submersibles meet this requirement. Roll and p i t ch a re s m a l l , except in the region of resonance. Heave, o r v e r t i c a l motion, response has a hump i n important short-period waves, which contain most energy when i r regu la r seas a r e considered. Therefore, minimization of t h i s hump w i l l d i r e c t l y improve performance; a f a c t which can be substantiated by optimization of the dimensions and cross-section of the sub-

The motions of column-stabilized semisub- mersible platforms could formerly o d y be pre- dicted by performing model experiments, e i t h e r i n confused i r r e @ a ~ - seas o r i n r e m m

merged lower hulls . I n t h i s paper the influ- ence of the lower-hull cross-section on the v e r t i c a l motions of f l o a t e r s w i l l be considered and discussed.

I n previous years the motions of ocean platforms could only be predicted from model t e s t s ; theory now permits approximation of t h e i r behavior.

A t N.S.M.B. a calculation method has been developed, which enables the estimation of wave exci ta t ion forces and motions .l The method assumes t h a t t h e platform can be s p l i t i n t o representative elements f o r which the wave exci ta t ion forces a re calculated. Addi- t i o n r e s u l t s i n the t o t a l force experienced by the platform from which i ts motions follow. These l a t t e r check well t h motion measure- 9 ments i n the model basin.

I n t h i s paper the fundamentals of t h e theory w i l l be checked against model t e s t re- sults of simplified platforms. The influence of the cross-section of various lower h u l l s on the v e r t i c a l motion w i l l be considered f o r the case of beam seas. Par t of the model t e s t r e s u l t s under discussion has been published.3 The knowledge derived from t h i s comparison w i l l be applied t o a comparative study between an

I r regular wave t e s t r e s u l t s give d i r e c t information about what i s t o be expected when the ful l -s ize platform operates under i d e n t i c a l circumstances. From regular wave model t e s t s , the response of the s t ructure t o each individual wave i s obtained which may be used f o r c o m p d - son purposes of various platforms or t o predic t the behavior in i r regu la r seas. Therefore a s t a t i s t i c a l method introduced by St. Denis and pierson4 f o r ship-motion theory i s used. Two assumptions are made: (1) an i r regu la r sea i s composed of an a r b i t r a r y number of regular s ine waves, each with d i f fe ren t length and height, and with random phase (pr inciple of superposi- t ion) and (2) motions i n regular waves a r e defined by amplitude and phase with respect t o the wave. When wave height increases, the motion w i l l increase l i n e a r l y i n proportion, whereas i t s phase w i l l be not changed ( l i n e a r i t y pr inciple) .

I f an i r regu la r sea can be decomposed t n t o a large number of regular waves, the response of a semisubmersible can a l so be assumed t o be cos- prised of the response t o each of these regular waves. Thus t h e s t a t i s t i c a l method f o r isregu- l a r seas can be used f o r predicting i r regu la r motion character is t ics . Performing model t e s t s f o r optimization purposes i s cost ly and time consuming, Nevertheless, optimization is, in most cases, a must since the re i s a lack of s t a t i s t i c a l da ta about the influence of the underwater h u l l shape upon platform motions. Therefore, mathematical methods have recent ly been developed t o predic t motions of f loa t ing structures.

F u j i i et al .5 prepared computer programs f o r t h e calculation of the hydrodynamic forces act ing upon marine s t ructures , using experimen- t a l l y obtained added mass and damping.

Ochi e t al.6 and Tasai e t al.? have pre- sented a method t o estimate wave forces using a s t r i p theory. The method used i n the present paper was developed by ~ o o f t l assuming t h e following .

1. The f loa t ing s t ructure can be subdi- vided i n t o s m a l l elements, f o r each of which t h e wave-excited and hydrodynamic forces a r e deter- mined.

exist& semisubmer~ible and a l tkrnat ive '

designs, with various lower-hull cross-section. I n pa r t i cu la r , the heave motions wi l l be re- viewed f o r regular waves a s well a s f o r i r r e - gular seas, which are representative f o r two

2. Calculation of these quan t i t i e s i s based upon p o t e n t i d theory fo r ' s m a l l bodies. Thus the f l u i d i s inviscid, incompressibLe and irrotationd.

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0% 1627 H. L. MINKENBEEG and M. F. van SLUIJS 11-133 t

3. Each element i s not affected by the presence of others.

4. Added mass and the mass coupling co- e f f i c i e n t s a re only influenced by t h e cross- sect ion and shape of the body and not by the

mably discussed in t h i s paper, i s described by

2 d Z ( I ) V + m y z ) = + N~~ ' C Z Z z *

d t 2

F,, COS ( w t + E ~ ~ ~ ) . . . . . . . . . (2) frequency of o s c i l l a t i o n which only holds t r u e a t low frequeneies.i, 7

5. The e f f e c t of damping i s small outside the resonance region and can be neglected.2 A t resonance, the damping can be taken Pto account by using Hooftls approximation.

6. The wave-excited forces a re dLvided i n t o an undisturbed wave-pressure force (~roude-Kriloff hypothesis) and an i n e r t i a force, being i n phase with the wave o r b i t a l acceleration. Fundamentally, the contribution of t h e dampin t o t h e wave-excited force should 5 be inc1uded;l a contribution which is, how- ever, s m a l l and can henceforth be neglected.l t7

I n order t o be informed of the r e s u l t s of t h e various calcula t ion methods, Fig. 1 has been compiled, where a comparison i s given with experimental data. It shows the v e r t i c a l wave force act ing upon a f l o a t , consist ing of a sphere attached t o a v e r t i c a l cylinder. Ochi uses s t r i p theory and takes i n t o account t h a t damping and added mass a r e dependent upon frequency. Hooft takes t h e added mass f r e e quency independently and approximates it with t h e following relat ion:

3 1 = ' K R p - ( ' + ~ ' " ~ ) f ' ( l ) 3 4

where mZ, = added mass in v e r t i c a l d i rec t ion R = sphere radius P = mass density of f l u i d a! = Fig. 1

Eq. 1 i s derived from po ten t ia l theory, ignoring t h e wave pressure contribution of t h e in te r sec t ion between sphere and column. Fig. 1 c lea r ly shows t h a t differences up t o 25 percent may r e s u l t i n t h a t frequency region where most waves carry energy, viz, CJ > 0.5 rad./sec. Generally, Ochi overestimates the v e r t i c a l wave-excited forces f o r t h i s case, whereas Hooft's r e s u l t s l i e within t h e r e s u l t s of t h e experiments conducted by Motora e t and Mercier. 9

OF LOWER-HULLS IN WAVES

From Newtonls law of dynamics, it follows t h a t the motions of a r i g i d body of a r b i t r a r y shape can be described by s i x coupled d i f fe r - e n t i a l equations of t h e second order with con- s t a n t coeff ic ients ; see f o r instance Refs. 10 and 11. If the semisubmersibLe i s assumed symmetrical i n longi tudinal and t ransverse direction, the vertical motion, which will be

The v e r t i c a l wave-excited force, F,, cos f & t

* E F J ) , can be generally wri t ten a s

F,, C05 ( w t + € F , c ) - CL1 m z z -

d t 2

where 61, 112 and p3 are correction f a c t o r s f o r t h e v e r t i c a l wave o r b i t a l motion, which can be deducted from the veloci ty po ten t i a l of the undisturbed wave.

When t h e damping influence i s neglected, Eqs. and can be

( p V + m z z ) - d z + C,, z =

d t 2

F,, C O S ( W t + E F z b ), . . . . . . . . . (4)

and

d 2 5 F,, c o s ( W t + EFZ 5 ) = 111 r n ~ z 7 + P g C z z c .

d t

. . . . , . . . . . . . . . . . . . . ( 5)

The solution of Eq. L, can be wri t ten a s

z = Z, c o s c w t + E ~ L ) - . . . . . . . . ( 6 )

s u b s t i t u t i o n o f Eq. 6 i n t o E q . 4 g i v e s Fz a

Z a L a , . 4 - ( 7 ) 5, 1 c z z - ( P V + ~ Z Z ) w 2 1

which i s t h e heave response outside of reson- ance. Resonance response should be estimated by introduction of the hydrodynamic damping, for the theory does not take it into account. The hflrodynamic damping paten- tial. damping, which is proportional to the velocity of oscillation and (2) viscous damping, which i s proportional t o t h e veloci ty of o s c i l l a t i o n squared. Since t h e viscous damping dominates the po ten t i a l one, damping

primarily depend upon platform motions and thus upon wave height. A t resonance, t h e hydrodynamic mass force cancels the res tor ing force. Therefore, the energy i n t h e v e r t i c a l wave-excited force can only be absorbed by t h e viscous damping force; Eq. 2 r e s u l t s i n

a.,, 1 = F z , c ~ ~ ( W Z t + E F z ~ ) . . . (8)

The t o t a l energy over a wave period a t resonance is

T l'

fizz 121% -$- d t = / F ~ ~ c o s I w t

0 0

stsgdt031
高亮
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I l

From Eqs. 6 and 9 follows heave response a t resonance:

The ver t ica l wave-excited force amplitude, Fza, and the viscous damping coefficient, Qzzl are calculated i n accordance with Ref. 1.

A r e a l i s t i c value of 2.5 m f o r wave ampli- tude was taken, which was used f o r comparison purposes between the various lower-hull shapes so as t o maintain consistency. The lower hul ls considered, are connected t o two circular cylindrical surface-piercing columns with a diameter of 6 m. The over-all length i s 60 m. The distance between column centerline i s 45 m. Displacements of 2,520 cu m and 2,990 cu m are considered, corresponding t o draf t s of l 9 and 27 m., respectively, measured t o the centerline of the lower hulls. The lower hul l cross- section was varLed as follows.

Variant I - @ -. 6.0 m diameter I Variant 11 - E! - 5.3 X 5.3 m I Variant 111 - @ - 8.0 X 4.0 m

Variant I V - @ - 4.0 X 8.0 m.

For these configurations, model t e s t s were conducted by Wahab and Van ~ l u i j s 3 i n the Seakeeping Laboratory of the Netherlands Ship Model Basin. Model dimensions and the t e s t arrangement are shown i n Fig. 2.

The model data plots are compared with the resu l t s of the computation tha t are shown in Figs. 3 through 10. On the whole, a satisfac- tory agreement eAs t s . Differences, origi- nating only a t minimum heave frequency, may be attributed t o the following three reasons.

1. Due t o l i t t l e heave motions, model measurement resu l t s must be interpreted with reserve f o r this frequency.

2. Theory ignores damping influence on the wave and hydrodynamic forces because of i t s insignificant effect outside of resonance. A t minimum heave response, however, where added mass cancels the pressure part of the wave force, motions are solely determined by the damping, which i s , as ea r l i e r stipulated, neglected by theory.

3. Theory has overestimated the added mass in part icular cases, which makes minimum heave response sh i f t t o lower wave frequencies. Table 1 gives a comparison of natural heave

periods and added mass, computed and derived from the model measurements. The theoret ical natural periods follow from Eq. 7.

Model natural heave periods were dertved from motion decay records. Added mass of the vari- ous configurations was calculated by substitu- t ion of the measured natural period, T , in to Eq. 11. It appears from Table 1 tha t goth the calculated and model values of added mass of Variants I1 and I V agree well. Added mass of Variants I and I11 are apparently overestimated by theory, which resu l t s i n a s h i f t of reso- nance and minimum heave t o longer waves. This i s clearly demonstrated by Figs. 3, 5, 7 and 9.

For deeper draf ts , ver t ica l wave forces w i l l be obviously reduced, a fact , which d i rec t ly follows from Eq. 3. Hence, heave response also decreases outside of resonance fo r deeper drafts, when added mass and re- storing forces are assumed t o be constant,

Eq. 11 suggests tha t longer natural heave periods can be attained a t larger displace- ment~, which i s confirmed by the data given i n Table 1. This does not mean, however, tha t heave w i l l also be l e s s a t resonance fo r deeper d r d t , because wave forces a t longer periods can be larger. Nevertheless, a comparison of Figs. 3 t o 6 with Figs. 8 t o 10, indicates tha t , in general, heave response decreases when draf t i s increased. From Eq. 5 it follows tha t the heave force can be s p l i t in a Frode- Kriloff component dependent on the wave eleva- t ion, and an added mass component tha t depends upon the wave orb i ta l acceleration.

I f the wave elevation varies a s = l d2t ; L, COS w t , then -=-w25 . Thus Eq. 5

becomes d t

The ver t ica l wave force w i l l be zero f o r one part icular frequency. This excitationless frequency i s

It depends only upon the waterplane area and the added mass. For the investigated con- figurations, having a constant waterplane area, the effect of lower-hull section on the wave- excited force i s i l lus t ra ted in Fig, 11. The ver t ica l wave-excited force i s considerably lower f o r the ver t ica l e l l i p t i c a l section,

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OTC 1627 H. L. MINKENBERG and M. 'F. van SLUIJS 11-135

I I owing t o i t s Lower added mass, a t wave fre- quencies higher than 0.5 rad,/sec. Fig. 12 demonstrates tha t then heave i s low also. V a r i a n t 111, with a horizontal e l l i p t i c a l sec- t ion, experiences the highest wave forces due t o i ts largest added mass. However, fo r wave frequencies smaller than 0.35 rad./sec, the wave force i s smallest. Thus, f o r waves with

> 0.5 rad./sec, heave response i s largest fo r Variant III ( ~ i g . 12).

A t resonance, heave i s predicted from the ver t ica l wave force and the damping, as indi- cated by Ref. 1, The ver t ica l wave force was approximately the same f o r the configurations considered, Variant I V has l i t t l e damping a t resonance, while the damping fo r Variant I1 i s large, which i s confirmed by the model t e s t results. For semisubmersibles the resonance heave frequency i s of importance f o r operation in irregular seas, For instance, the res onse of Variant I V t o waves with w > 0.5 rad .zec i lowest. Due t o a small added mass, however, the resonance frequency l i e s in the range of actual wave frequencies; also i t s response a t resonance i s large. Therefore, 1 p g e heave motions wi l l be encountered in irregular seas with a large average period, t h i s in contrast t o the behavior of Variants II and 111,

MOTIONS OF SINGLE LOWER HULLS I N IRREGU SEAS

Based upon the superposition and l i nea r i t y principle, the significant motfons of a plat- form in irregular seas can be predicted from the response t o regular sine waves, The i r regular seas are assumed t o be long-crested, i n which case t h e b energy spectra can be described by

where I: 2

S ( W ) = d w 45.

L do,

Apart from the values of A and B, Eq. l 4 depicts spectra similar i n shape t o the pierson-~oskowitzl~ formulation f o r f u l l y developed seas.

When the wave energy spectrum i s related t o actual observations a t sea, it follows tha t the average observed wave height conforms generally t o the calculated mean of the one- t h i rd highest waves (= significant wave height) and tha t the observed period corresponds t o the calculated mean wave period, Thus f o r a narrow band spectrum

observed wave height = I

observed wave period = 9

From Eqs. 1-5, 16 and 14, it follows tha t CO- e f f ic ien ts A and B are related t o significant wave height and mean wave period.

The values of significant wave height and mean period used apply t o conditions a t the North Sea and south of Australia. They were taken from information gathered by ~ e t r i l 3 and Hogben and Lumb. l k The m a i n difference of weather in these areas i s tha t dominant periods of the waves a t the North Sea are shorter than those a t South Australia. The significan& wave height was held the same fo r both areas. The spectra as formulated above are i l l u s t r a t ed by Figs. 13 and 14.

The resu l t s of the i r regular heave calcu- la t ions fo r the various lower-hull configura- t ions are given i n Fig. 15. To t h i s purpose, use was made of responses given in Fig. 12 a s follows :

Eq. l 9 i s the sigrdficant heave motion, double amplitude, which i s the mean of the one-third. highest up and down values.

As the North Sea, thus i n relat ively short waves, the ver t ica l e l l i p t i c a l cross section oi the lower hul l wi l l be favorable up t o wave heights of about 4 m. I n rougher seas heave increases considerably. The horizontal -p- t i c a l cross-section i s the most unfavorable regarding heave, while the difference of heave motion between the circular and square cross- section i s negligibly small.

For the South Australia area, an opposite trend appears tha t i s caused by the larger amount of energy i n the long waves of the spectrum having frequencies i n the v5cinity of heave resonance. This resu l t s i n large verti- ca l motions fo r the vertical, e l l i p t i c a l cross- section, the more so a s i t s damping i s very small.

Summarizing, f o r sea areas with re la t ive ly short wave periods such a s the North Sea, Con- figuration I V with a ver t ica l e l l i p t i c a l cross- section wi l l heave less , whereas i n longer I

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11 4-36 MOTION OPTIMIZATION OF ~SUBMERS1BL;ES OTC 1627 I I l

waves such a s the South Australian area, preference must be given t o the square o r t h e horizontal e l l i p t i c a l cross-section.

L a HULL OPTCMIZATION FOR A SENISUENERSIBLE

The howledge gained from t h e study about the e f fec t of lower-hull sect ional shape upon the v e r t i c a l motion in beam waves w i l l be adopted t o optimize heave f o r an ac tua l semi- submersible. The semisubmersible under con- s idera t ion i s intended t o be used f o r laying pipes on the seabed. It consis ts of two sub- merged longi tudinal c i r c u l a r cyl indr ical hul ls , each of wkich i s connected t o f i v e surface- piercing v e r t i c a l columns f o r s t a b i l i t y . Deck and h u l l s a re connected by bracings such t h a t a f u l l symmetry e x i s t s with regard t o a v e r t i c a l plane through the center l ine and through the midship section. M a i n par t i cu la r s a r e given in Table 2 f o r an operating d r a f t of 63 f t .

This barge has been used by Van S l u i j s and an^ f o r making t h e i r experimental and theore t i ca l motion corre la t ion, For the optimization study, two lower-hull sections were chosen; being the v e r t i c a l e l l i p t i c a l shape which i s favorable at t h e North Sea and the square section, which i s a r e a l i s t i c a l t e rna t ive f o r t h e North Sea a s well a s the South Austra l ia area. The shape of t h e lower- h u l l sections was a l t e red while t h e area had been kept constant and equal. t o the o r ig ina l design. Character is t ics of the weight d i s t r i - bution were kept constant throughout;. I n Figs. 16 and 17 t h e calculated heave and r o l l response t o regular beam waves a re shown.

For wave frequencies l a r g e r than 0.5 rad./ sec, heave,response of Alternative I1 i s con- siderably smaller with respect t o t h a t of t h e other configurations. Due t o a s m a l l added mass, about 50 percent of t h a t of the o r ig ina l design, resonance occurs i n t h e area of t h e longer waves. This pa r t i cu la r section shape has l i t t l e damping, which r e s u l t s i n l a rge response a t resonance estimated by Eq. 10 f o r a wave height of 5 m. The magnitude of these values might be a r b i t r a r y f o r comp&son purposes; however, they serve well.

I n short period waves, the difference i n heave between the o r ig ina l design and Alterna- t i v e I a re small. The response a t resonance of Alternative I i s smaller, which i s mainly a t t r ibu ted t o a Large viscous damping of t h e square cross-sections. Also i t s added mass i s somewhat l a rger , r e su l t ing i n a longer na tu ra l heave period. Calculated na tu ra l periods f o r heave and r o l l a re shown i n Table 3.

model experiments.2 Roll response i s shown i n Fig. 17. Apparently Alternative I1 r o l l s l e s s , which must primarily be ascribed t o a difference of s t ab i l i ty . For the e l l i .p t ica l shape of the lower hul ls , the d r a f t was 3.4 m deeper, r e su l t ing i n a lower posi t ion of t h e center of buoyancy while, a s already said, the center of gravi ty was held constant.

Since the waterplane area and t h e dis- placement are t h e same f o r a l l corrfigurations, t h e height of the metacenter above t h e center of buoyancy remains constant. This makes t h e metacentric height of Alternative I1 lower, r esu l t ing in a smaller r o l l res tor ing force, a l a r g e r r o l l period and a lower response t o high-frequency waves. The difference in r o l l charac te r i s t i c s between the o r ig ina l vessel am Alternative I i s negligible. Their s t a b i l i t y i s t h e same.

I r regu la r heave i s given in F'ig, 18. It shows the same trend a s was observed f o r t h e s ingle lower h u l l motion calculatFon. With North Sea waves of up t o 4 m, a v e r t i c a l e l l i p t i c a l shape i s superior. I n higher seas t h e v e r t i c a l motion w i l l increase appreciably since t h e longer wave components exci te the platform in i t s na tu ra l period.

No p r a c t i c a l preference regarding heave e r d s t s between t h e o r ig ina l design and its Alternative I f o r the investigated North Sea conditions, For the South Australian area a square lower-hull shape i s favorable; a v e r t i c a l e l l i p t i c a l section or iginates con- s iderable heaving.

From Fig. 19 it follows t h a t the smallest r o l l angles a re experienced by Alternative I1 because of i t s s t a b i l i t y data. The difference h r o l l i n g charac te r i s t i c s f o r t h e or2ginal semisubmersible and Alternative 1 i s hsignifi- cant. The magnitude of r o l l i n g f o r a l l con- f igurat ions i s considerably l e s s than t h a t of shg le -hu l l conventional barges, a f a c t which i s one of the great advantages in using semi- submersibles f o r operations t h a t a r e t o be performed when on s ta t ion.

CONCLUDING REMARKS

I n the paper a comparison has been made between v e r t i c a l motion data derived from model experiments on s ingle lower h u l l s of various cross-sections and those calculated by a l inear ized theory, It i s shown t h a t a sa t i s fac to ry agreement e l d s t s between t h e measured and calculated motions so t h a t t h e fundamentals, upon which the theory i s based, a re sound.

The computed values f o r the o r ig ina l de- sign agree very well with the da ta from the

The e f fec t of va r ia t ion of t h e lower h a cross-section shape on the motion 1

I

Page 7: Motion Optimization of Semi-Submersibles

0% 1627 H. L. MI-G and M. F. van SLUIJS 11-13?

characteristics of an actual semisubmersible has been investigated t o detect trends leadlng t o optimum performance under different environ

I mental conditions. I n t h i s respect, the added mass turns out t o be the ruling factor.

Theory i s f e l t t o be useful t o calculate platform motions fo r various design con- figurations t o arrive a t the optimum shape f o r i t s specific operation. In addition, model t e s t s w i l l be necessary t o check the theoreti- ca l predictfon and t o obtain information on phenomena tha t are unpredictable by calcula- tion.

NOMENCLATURE

C,, = restoring force coefficient i n v e r t i c a l direction

F,, = ver t ica l wave-excited force amplitude g = acceleration due t o gravity 1

k = 2 f t = h

wave number I kLa = wave slope amplitude mzz = added mass i n ver t ica l direction NZ, = potential damping coefficient in

ver t ica l direction Qzz = viscous damping coefficient in verti .

cal direct ion R = sphere radius

S g (W) = wave spectral density T = wave period S

T = mean wave period TZ = natural heave period

Z = ver t ica l displacement Za = heave amplitude

2GIl3 = significant heave motion (double amplitude)

= phase angle between ver t ica l wave- ' 5 excited force and wave

phase angle between heave motion and ' "= wave

X = wave length

w t 2 3 = wave circular frequency T I

ao = frequency f o r minimum wave-excited I 2 E

force u'2=F = natural heave frequency

p = mass density 5 = wave elevation

6, = wave amplitude cW = significant wave height

4, = r o l l amplitude PTa, = significant r o l l motion (double

amplitude) C L ~ , ~ - L ~ , C L ~ = correction factors f o r ver t ica l wave

orb i ta l motion V = displacement volume.

1, Hooft, J. P. : "A Mathematical Method of Determining Hydrodynamically Induced Forces on a Semi~ubmersible~~, paper presented a t the SNAME Annual Meeting, NOV. 11-12, 1971.

2. van Slui js , M. F. and Tan, Seng Gie: "Experimental. and Theoretical Motion Correlation of a Pipe-Laag Barge", Publication 375, N.S.M.B., symposium on Offshore Hydrodynamics, Wageningen, Aug. 25-26, 1971.

3. Wahab, R. and van Slui js , M. F.: "Some Remarks on Model Tests ~ 5 t h Floatirw Platforms in Waves", Marine ~echnolom ( ~ c t . , 1968) 5, No. 4.

4. St. is, M.-&I~ Pierson, W. J.: "On the Motions of Ships i n Confused Seas", Trans., SNAME (19%) 61.

5. hjii, Hitoshi and Tarahaski, Takeski: l1Estimation of Hydrodynamical. Forces Acting on a Marine Structure", Technical

eview, Mitzubiski Heavy Industries, Ltd. May, 1970). P-

6. Ochi, M. K. and Vuolo, R. M, : "Seakeeping Characteristics of a M u l t i - U n i t Ocean Platform", paper presented a t the SE- Spring Meeting, May 25-28, 1971.

7. Tasai, F., Arakawa, H. and Kurihara, M,: "A Study on the Motions of a Semi- Submersible Catamaran H u l l in Regular Waves", Reports of Research Ins t i t u t e f o r Applied Mechanics (1970) X V I I I , Ho. 60.

8. Motora, S. and Kogama, T. : "On Wave ~ x c i t a t i o n l e s s ship ~ o r m s ~ ~ , J.S.N.A., Japan (1969) 3.

9. ~ e r c i e r , J. .A:: llHydr~dynamic Character- i s t i c s of Several ert tic-al Floats in Wavesll, Report SIT-DL-70-1481 Stevens Ins t i t u t e of Technology (1970).

10. V u g t s , J. H. : "The Hydrodynamic Forces and Ship Motions in Waves",. thesis , Delft U. of Technology (1970)..

11. Hanaoka, T., e t al. : "Researches on Sea- keeping Qual i t ies of Ships i n Japan", 60th Anniversary Series (1963) 2.

12. Pierson, W. J , and Moskowitz, L.: "A Proposed Spectral Form f o r N l y Developed Wind Seas Based on Similar i ts Theom of S. A. Kitagarodskiifl, Jour. keophys: Res. ( ~ e c . , 1964) &.

13. Petr i , 0.: l lStatistik der Meereswellen i n der Nordseell, Einzelverof f entlichung Nr. 17, Deutsche Wetterdienst, Seewetteramt, Hamburg.

1.4. Hogben, N. and Lumb, F. E. : "Ocean Wave Stat is t ics" , National Physical Laboratory, Teddington.

Page 8: Motion Optimization of Semi-Submersibles

TABLE I - COMPARISON BETWEEN CALCULATED AND MEASURED ADIlED MASS AND NATURAL PERIOD OF HEAVE

TABLE 11 - MAIN PARTICUIARS OF SEMI-SUBMERSIBLE BARGE

I Draught

I 19.0 m

I

,TABLE I11 - NATUBAL PERIODS OF THX SEMI-SUBMERSIBLE

, 27.0 m 1 Variant I - Q 1 172.8 1 'l::: 1 'l::: 1 Variant I1 - 204.6 189 5 Variant I11 -a 304.2 277 W 5 20.4 20.0

I Variant I V - @ 76.2 16.2 16.2

!?atufal period i n sec.

liatural period

Configuration

Variant I Q Variant E1 - H Variant I11 -a Variant I V - @

Calculated

17.2

17.9 19.7 15.1

1 Magnitude

400

160

63 30,118

45.4

13.9

68.0 1

l Designation

Length

I izzt even keel Displacement weight

I Center of gravity above base

I Transverse metacentric height

I Kadius of gyration h

r o l l d i rec t ion

Original

Keasured

16.7

17 7 l 9 . l

15.3

heave 16.1 (15.6)

27.7 (2q.9)

Added mass i n ton. mnaec?

Symbol

L B

T

V

In

m

k 4

i J t e rna t ive I heave 16.8 1 0 1 1 I, 1 28.8

. Calculated

172.8

204.6

304.2

76.2

Unit

f t . f t . f t .

sh. tons

f t .

f t .

f t .

I1 heave 1%.2

0 1 1 1 3 . 6

Measured

147.0

197.0

273.5 81 .O

( . . . . ) Measured values, derived from [2] . .

Page 9: Motion Optimization of Semi-Submersibles

-.-.I S.MOTWA

--W M K. OCHl

I l HOOFT ] cALCuLATEO

FREOUENCY in n d .

F i g . 1 - C a l c u l a t e d and measured wave- e x c i t e d v e r t i c a l f l o a t e r f o r c e s .

l dlmrnrlonr in m l

4

3

2 -

U

;i-

"5

FRONT VlFW

.--L

1.20

CROSS SECTION OF -R-HULL A

VARIANT I - @ - 6 OlZOm VARIANT U - - 01063 x 01063 m

VARIANT m - e - OlsQ2 xa0796m VARIANT a - e - oomsxolsoa m

F i g . 2 - Mode l d imens ions and t e s t a r rangement f o r s i n g l e l o w e r - h u l l s .

T=100m

FREOUENCY in "d. FREOUENCY In rad sec.'

P

. 1-

FREOUENCY in rrd sec-1

W

VARIANT I @

CALCULATE0

----C--- MEASURED

I I l I I I I

l +

1 I

F i g . 3 - Heave response o f c i r c u l a r l o w e r F i g . 4 - Heave response o f squa re l o w e r F i g . 5 - Heave response o f h o r i z o n t a l h u l l , s h a l l o w d r a f t . h u l l , s h a l l o w d r a f t . e l l i p t i c a l l o w e r h u l l , s h a l l o w d r a f t .

p---- 0.5 ID

Page 10: Motion Optimization of Semi-Submersibles

FREOUENCV ~n rad. sec;' FREQUENCY In rad sec;'

4

3

2.

F i g . 6 - Heave response o f v e r t i o a l F i g . 7 - Heave response o f c i r c u l a r e l l i p t i c a l l o w e r h u l l , s h a l l o w d r a f t . l o w e r h u l l , deep d r a f t .

T=100m "

F i g . 9 - Heave response o f h o r i z o n t a l e l l i p t i c a l l o w e r h u l l , deep d r a f t .

VARIANT H @

CALCULATED

---- MEASURED

-. - .

4

3

2

I

-----

L"

'j-

FREQUENCI in "d.

F i g . 8 - Heave response o f squa re l o w e r h u l l , deep d r a f t .

1

OO O S 1.0 FREQUENCY in rad.$ec.-l

T P70m

FREOUENCV in rad.s.c:'

VARIANT m @

CALCULATED

------ MEASURED

F i g . 10 - Heave response o f v e r t i c a l e l l i p t i c a l l o w e r h u l l , deep d r a f t .

Page 11: Motion Optimization of Semi-Submersibles

FREQUENCY in r.d.sec:l

T-l9Om

- VARIANT I e ---- VARIANT n m --- VARIANT m e

VARIANT X 8)

FREOUENCY in rrd %+c:'

4.

3

2 -

F i g . 11 - C a l c u l a t e d w a v e - e x c i t e d v e r t i c a l F i g . 12 - Heave r e s p o n s e o f t h e v a r i o u s F i g . 13 - N o r t h Sea wave spectra. f o r o e o t t h e v a r i o u s l o w e r h u l l s , l o w e r h u l l s , s h a l l o w d r a f t .

s h a l l o w d r a f t .

a i !

--

,/-'-.N. 0 05 1.0 1.5

FREQUENCY in n d sec-'

T S 19.0m

FREQUENCY in rad, rcc;l

F i g . 14 -. S o u t h A u s t r a l i a n wave s p e o t r a .

WRIANT I e ---- --- VARIANT U

--p VARIANT m o

4

3

VARIANT I O

----- VARIANT U T i 1 9 O m --- WRIANT m. a l -

VARIANT H b

F i g . 15 - I r r e g u l a r heave o f t h e v a r i o u s l o w e r h u l l s , s h a l l o w d r a f t .

\, \ 2 I ".

SOUTH AUSTRALIA SEA STATE

A

B

C

0

C

LV31n m ( 7 in S=.

1.60 6.0

2.50 0 0

3 60 I 10.1

4.00 10.0

Page 12: Motion Optimization of Semi-Submersibles

ALTERNATIVE I

-.-.- ALTERNATIVE n @

. - FREOUENCY 10 rrd sec-l

Fig. 16 - Heave response o f original platforms and two alternatives.

- ORGINAL 0

-I- ALTERNATIVE I Q

Fig. 18:- lrregular heave of original platform and two alternatives.

ORGINAL

-- ALTERNATIVE 1

- ALTERNATIVE U @

FREOUENCY in rad. rcs"

F i g . 17 - Roll response of original platform and two alternatives.

ORGINAL

-.-- ALTERNATIVE n @

Fig. 19 - lrregular roll o f original platform and two alternatives.