Study on motions of floating bodies under composite external loads

Study on Motions of Floating Bodies under Composite External Loads

In the field of ocean development, various types of floating bodies are being designed and constructed. In most cases, motions of those bodies are under the effect not only of ocean waves but also of forces exErted by external systems such as mooring lines, pipe-line or cable undér laying, or other floating bodies connected together.

Extensive studies were carried out on motions of floating bodies under composite external loads and widely applicable prediction method was developed in a highly unified form for the relevant problems.

In order to verify the applicability of the method, model experiments were carried out for severaltypïcal cases of floating bodies under composite external loads. Fairly good agreement was obtained between calculated and measured values, and availability of present method was confirmed.

As far as the composite external loads can be approximated by linearized dynamic system present method can widely be applied to the practical design problems of various types of floating bodies under composite external loads

Introduction

In the field of ocean development, various types of float-ing bodies have been designed and in operation. For the design of such bodies, it is essentially important to predict their dynamical behaviours in waves. Their motions havé to be treated taking account not only of environmental forces such as thosé due to waves and winds, but also of forces acting on them due to the presence of external systems such as mooring lines, pipe-line or cable under laying, or connected floating bodiés. The latter forces which are dif-ferent from environmental forces are called the "composite external loads" in the present study. In general, the dy-namical behaviours of those floating bodies under such

external loads are extremely complex.

In recent years, many studies have been carried out in regard to the motions of floating bodies in waves under the composite external loads. Most of them are for some particular kind of external loads; e.g., only fôr ordinary

mooring lines, or only fòr towed cables, etc. However, basic

patterns of the approach to those problems appears to be rather common. Motivation of the present study, therefore, has been to investigate the problems fromfundamental and systematic view-point, and to develop a widely applicable unified method to estimate the dynamical characteristics of floating bodies in waves under multifarious external loads(')(2).

In the present study, above-mentioned unified predic-tion method was developed for linear mopredic-tions of floating bodies in regular waves under multifarious external loads, based on the strip method widely used for the prediction of ship motiOns. Several modal tests in waves were carried out for typical cases of moored or connected floating bodies, and applicability of the present method has been confirmed. Present paper describes an outline of the study.

PredictiOn method of motions in waves

Basic considerations are made on a floating body system composed of multiple bodies connected and moored as a common configuration of floating bodies under composite

external loads. The characteristics of external loads in gen-eral are non-linear. In many cases, howEver, thEy can be treated as a linear system on assumption that amplitude of waves and motiOns of floating bodies are sufficiently small compared with their main dimensions. it is assumed here also that oscillatory motions of a body in waves are linear and harmonic, and connecting or mooring members are of negligible mass and also are treated as a linear dynamic

system.

In the prediction of motions of multiple connected bodies, not only mutual actions between bodies through connecting members but also hydrodynamic interactions must be taken into consideratiOn. lt can be understood that the mutual interaction due to connecting members is a "mechanical connection", and hydrodynamic interaction is a "hydrodynamic connection". Therefore, both effects can be analyzed essentially by the same way in a unified ana-lytical formulation of motions of the floating bodies.

Kunihiro Ikegami* Masami Matsuura*

Fig. i Definition of

six-degree-of-free-dom motiòn

2.1 Equation of motions

Analytical procedure for prediction of motions ofa body freely floating in waves is thé basis of the present problem, and for that procedure the strip method(3) is adopted. The six-degree-of-freedom motions are definedas shown in Fig. 1. On assumption that the motions_{are linear} and harmonic, the equation of motions for _{a single body} freely floating in waves is given as follows in matrix

nota-tions,

(M+A)1+B+Cfl=F

(2.1)

X

Body-I Fig. 2 Coordinate

systemför °

a moored Fig. 3 Coordinate system for connected

body bodies

where the element K11 of the matrix K is the i4h mode force acting on the mooring member obtained from the j-th mode forced oscillation of the member jointed to â body at the point 0.

Defining the transformation matrix T (see AppenLlixA) from the cootdinate system o-xyz to 0-îj71 in Fig. 12, the added coefficients matrix G due to a mooring member are

gven as follows:

G=TtKT (2.6)

where the Superscript t is referred to the transpose! of the matrix.

Next, it is supposed that the two bodies are connected together at the points O and 0m as shown in Fig. 3. It i assumed that the transformation matrices T, and Tm are defined for the transformatiOn from the coordihte sys-tems OFX,Y,z, to OrÎ,Y,i, and from to °m

m9m Im, respectively, where in the coordinate systems örYzzz Bfld 0mmYmm, the one is a paiállel trañslation of the other. SirnThirly fór the case of a moored body, the

added coefficients E are expressed as follows

E11 TfKT,

EimTîKTm (1m)

(2.7)

where the matrix K describes the characteristics of a con

necting member.

it is assumed that forces and moments acting on the body àt the mooring or connecting point aïe cained not only by the deflection of mooring or connecting member but also by velocity and acceleration of the deflection. By taking into account the damping and the inertia forces as well as the restoring forces, the matrix K can bewiritten in

the same form as the matrixD shown in Eq. (2.2). Its form

can be written as follows,

K=e2AK +iBK +CK

(2.8).

where

AK : mass coefficients due to inooïing or connecting

member; 6 x 6 matrix

BK damping coefficients due to mooring or con-necting membei;6 x 6 thatrix

CK : restoring coefficients due to mooring ór

con-necting member; 6 x 6 matrix

The elements of the matrix K representing the character-istics of mooring and connecting members can be written

as follows.

Kjj_C&)e24+iWeBj/4Cj/ ;i,ji,2,. ..,6 (2.9)

The element K1 is the i-th mode reaction force acting on the mooring and connecting point obtained from the /-th mode motiön of that point. This is defmed in reference

g

M generalized mass for body; 6 x 6 matrix

A : added mass; 6 x 6 matrix

B : damping coefficients; 6 x 6 matrix

C :

hydrostatic restoring coefficients; 6 x 6 thatrix

1) : translatory and angular displacements of body;

6 x ¡ vector

14

=Èe[,1eet] ;

_j=1,2, -.

6

liAi: complex amplitude

F : wave exciting force and moment: 6 x i vector

FI=Re[FA,e"éfl ;j=l,2,...,6

FAI: complex amplitude

wave circulár frequenc of encounter A: subscript indicating complex amplitude

j:

subscript indicating mode of motioñ,j=1,2, 6 corresponding to suige, sway, heave,

roll, pitch and yaw, respectively

Defining the linear operator D as follows:

DWe2(M+A)4iWeB+C

(2.2)

Eq. (2.1) may be simplified in the form:

D!ÌA=FA

(23)

Expanding Eq. (2.3), the equation of motions for multi-ple bodies moored and connected each other is given in a

general form as foïlows:

TD,+D;,E;;+G,

_J

Dm,,, +E,G,

(2.4)

1)1714 FmA+PrnA

where

D, : coefficientsfor a singlebody freely floating;

6 x 6 matrix

hydrodynamic coefficients due tO hydrody-namc interaction among multiple bodies;

6 x 6 matrix

G, : added coefficients due to mooring members;

6 x 6 matrix

Eim : added coefficients due to connecting mem-bers;6x6 matrix

FIA : wavè exciting forces for a single body freely

floating; 6 x 6 mâtrix

PIA: added wave exciting forces due to hydrody-nainic interaction among multiple bodies;

6 x 6 matrix

i, m : Subscripts indicating quantities related to

number of bodies

The added coefficiénts matrices D'im or Eim can be obtained from the forces acting on the 1-th body when the m-th body is forced to oscillate and the other bodies are restrained.

2.2 Added coefficients matrix due to mooring and

connecting members(4)

As shown in Fig 2, a floating body is assumed to be

moored at the point ö. The matrix K representing the

charactenstics of the mooring member is defined in refer

ence to the local coordinate system O-.Y.f as follOws,

to the local coordinate system in Figs. 2 and 3. The matrix K is a 6 x6 matrix in general, but it is reduced to

a 3x3

matrix when a member does not transmit moments as in the case for an ordinary mooring line. It is described as

follows,

where the matrix O is zero matrix. Coupling term K11 (ij) in general has a certain value such as the case for mooring line characterized by catenary theory. Determining the local coordinate system o-5Ï for mooring and connecting member suitably, however, coupling term can be neglected

for many a practical üse.

In case that the restriction condition at the bonnecting point should be satisfied such as for pin-joint connection and rigid connection, characteristic matrix can be described

by diagonal matrix as follows.

In this case, only restoring coefficient C11K is considered,

and the magnitude should be so large that natural fre-quency- of dynamic system becomes at least one order of magnitude larger than the wave circular frequency of en-counter.

Treating the mooring or connecting member as a linear dynamic system (mass-damper-spring system) in unified manner, the present method can be applied to wide variety types of motion problems- of floating bodies under com-posite external loads. However, there are some cases where linearization is difficult especially when mooring and con-necting members have strong non-linear property such as in the case for a rubber fender. Therefore, in order to estab-lish the linearization technique of such cases, model test was carried out and several linearization methods were

evaluated. The results are described in Appendix C. 2.3 Hydrodynarnic interaction among multiple bodies The approximate theoretical method for the calculation of hydrodynamic forces including interaction among multi-ple bodies was developed by Ohkusu(5)(7), which can be

partly applied to the present study. According to this

method, when the two-dimensional radiation problem for each body in the multiple bodies is solved beforehand, the

two-dimensional hydrodynamic forces can easily be derived

including the hydrodynamic interaction among multiple.

bodies.

Considering the two-body problem as shown in Fig. 4, the two-dimensional hydrodynarnic coefficients due to hydrodynamic interaction between two sections can be

described as f011ows,

X

Fig. 4 Coordinate system for calculatiOn

- of hydrodynamic interaction

Dfpg AR(L.R)jHje_I

DRf/pg AR(R+L)JHZj e'

DRLU pg AL(L..R_)jH1 e

Df/pg AL(RL)JHLje_

where

DRRIJ, DLRfJ, DRLiI, DLLII : two-dimensional

hydro-dynamic coefficiëflts due to hydrohydro-dynamic inter-action; for example, DLRJ means i-th mode hy-drodynamic force acting on L-section when R-sec-tion is forced to oscifiate in j-th mode with unit

amplitude.

AR(L..R)/, AR(R.L)I, AL(LR)j, AL(R.L)i : ampli-tude of incoming wave resulted from the exchange of reflected waves between the two sections; fOr example, AR(R...L)/ means the incoming wave acting upon L-section when R-section is forced to oscillate in j-th mode with unit amplitude. (see

Appendix B)

H1, Hj :

Kochin function; for example, H1 means the Kochin function of R-section for i-th mode motion.

p water density

g : acceleration of gravity

-k wave number

2F spacing between the centres of

sections-Amplitude of incoming wave resulted from the exchange

of reflected waves between the two sections can easily be calculated according to Ohkusu's method based on an approximate assumption that there exists an interaction only -between the two sections with respect to the progress-ing waves generated by the oscillation or the reflection of one section. The two-dimensional wave exciting forces due to hydrodynamic interaction between the two sections can be described as follows in the similar manneï to Eq. (2.12),

F1 = _pg AR (LR)Hj e1'1'

FL1 = pg AR(R.L) Hj1 eH1 e"}

where

F1, Fj1: two-dimensional i-th mode wave exciting force due to hydrodynamic interaction acting on

R -section and L-section, respectively

AR(L+R), AR(R.+L) : amplitude of wave progressing

toward the right and the left, respectively, be-tween two sections when incident wave with unit amplitude comes from the right (see Appendix B) Further, applying the strip method, the thiee-dimen-sional added hydrodynamic coefficieñts Djm and wave exciting forces FiA can be obtained which are-required for

MTB158 July1983 (2.12)

K=

K11 O

__p_---O K22 O O 0

3j_

'K44

IO

i O-O 0 K55 O O O K66 (2.11) K11 K21 31 K12 K22 K32 o K13 K23 K33 o o (2.10)

deriving the solutions of motiöns inclúdi.ng the effect of the hydrodynamic interaction.

2.4 Forces actiúg on mooring and connecting members

In the present method, the forces änd moments acting on mooring and connecting members can be calculated by use of the motions of floating bodies obtained by solving Eq. (2.4). The forces and moments acting on mooring member can be expressed as follows in reference to the local coordinate system ö-îj)2 in Fig. 2,

WA =KT7ÌA (2.14)

where

WA fôrces and moments acting. on mooring

member; 6 x 1 vector

The forces and moments acting on connecting member can be described as follows in reference to thelocal

coordi-nate system o1-,y in Fig. 3,

VLmA = K(Tm7lmA - TflA) (2.15)

where

VImA forces and moments acting on connecting

member between l-th body and m-th body

3. OÙtline of computation system

According to the present. theory,, the computer program Motion Analysis Program of Composite Floatmg Bodies

in Waves (MACOF)" has been developed. A brief outlineof

the computatibn system MACOF is presented in this sec-tion.

This total prögram system MACOF coñsists of four independent programs, MACOF-F, MACOF-I, MACOF-M and MACOF-O. The whole flow of computation systemis

shown in Fig. 5. The data each program are connected through the disc files. Each program is written in FORT-RAN iV and the whole system is of the order of about

/

Mooring or Connecting condition Wave direction"

/

Wave data Output control data MACOF -M Motion analysis for

composit floating bodies

MACOF.- O

Short- term prediction Output management

Fig. 5 Flow cha±t of computation

System MACOF

=

---,

Motion Mooring or connecting force_____ DISC FiLE-M Motion Motion Accelèration Relative motion Standard deviation 10 000 steps.

Functions f each prÒgram are outlined. as follows. MACOFF

The hydrodynamic coefficients and the wave exciting forces and mpmënts acting on each body of the composite floating bodies are computed by the strip theory. It is assumed that each body is lateral symmetric and freely floating independently in regular waves. Thisprogramihasa function solving the equation of motions also. The output data from the program are printed for the matrix of equa-tion of motiOns and the moequa-tions òf the independent body: The data of the hydrodynamic coefficients and the wave exthting forces and moments for each body are writtn on DISC FILE-F which will be used in MACOF-M. Th data of Kochin function for each strip section of the body are

written on DISC FILE-K which will be used in MACOF-I.

In this program two-dimensional hydrodynamic orces acting on each strip sectiòn can be calculated by two kinds

of numerical method shown as follows:

( 1) Multi-pole expansion-mappthg method with

Lewis-. form representation

This method is of most practical use because of the short computing time. The geometñc.l shape of each strip section is aimatic.1ly represented by the Lewis-form which has the same bearñ, diaft. and area as the given section. However this method is not applicable to the arbitrarj section shape being off the range of the

Lewis-fono representation. (2) Singularity distribution method

This method is time consuming but aôcurate for any Section shape. The user has the option to select it, if necessary. The geometrical shape of the strip section is represented by a given number of offset points.

3.2 MACOF-I

The hydrodynamic coefficients and the wave xciting

forces and moments due to the hydrodynamic interaction between the multiple floating bodies are computed They are calculated by use of Kochin functions stored dn DISC FILE-K according to Ohkusu's approxinate method. In the present condition tIns program is applicable to the two flOating bodies with an alongside configuratiOn at zero forward speed in beam waves. The calculated results aYe

written on DISC FILE-I Which wiil be used inMACÒF-M.

3.3 MACOF-M

This program is the major program of the present sys-tem.. Analysis of the motions in regular waves areavailable

for the cmposite floating bodies with "connected",

"moored" or any other composite configurations based on the theory outlined previously In this program the data for the hydrodynamic forces are read from DIS FILE-F

and DISC FILE-I produced by MACOF-F and MACOFI,

respectively.

The motions of each floating body including the moor-ing or connectmoor-ing forces and monents are printd-out in

terms of amplitudes and phase angles. Thecalculatd results

Of the motions are also written on the DISC FILEMwhich

wifi be used in MACOF-O.

3.4 MACOF-O I

This program computes the varIous responsefs such as

/

Principal dimension Hull geonletry Wave condition MACOFF Calculation for independent body IfydrodynamiC Motion force

DISC FILE-K DISC FILE-F Kochin fonction Hydrodynamic_torce

/

Body separation MACOF-I

Hydrodynamic iirthraction Wave condition Calculation for

hydrodynamic interaction

DISC FiLE-1 Hdrodynamic

relative motion and acceleration in regular and irregular waves by using the data on the motions of each floating body which were obtained by MACOF-M and stored on DISC FILE-M. The amplitudes and phases in regular waves and the standard deviations in short-term irregular waves are predicted for the following items:

Motions at the center of gravity of each floating body Motions at the specified points of eàch floating body Accelerations at the specified points of each floating, body

Relative motions with respect to wave surface at the specified points of each floating body

Relative motions between the floating bodies

Results obtained in this program can be printed-out as a table or shown graphically by the plotter.

4. Model test

In order to confirm the validity of present prediction method, model tests in waves were carried out for several kinds of moored or connected floating bodies in the Sea-keeping and Manoeuvring Basin of Nagasaki Experimental

Tank, Mitsubishi Heavy Industries, Ltd.

4.1 Floating body moored by linear spring

As a case of simplest external load, model test in waves was carried out with linear spring in vertical and in hori-zontal direction, respectively. Principal particulars Of tested model and test condition are shown in Fig. 6. Restoring moment due to spring was adjusted to 20 percent of hy-drostatic pitching moment in vertical direction, and 20 percent of rolling moment in horizontal direction,

respec-tively.

Case-1 Case-2

With spring in the horizontal With prin in the vértical

direction direction Waves Coil spring Waves Ship model : L = 4.2 m B = 0.698m A = 0.245m 4 = 576.7kg GM= 0.091m

Fig. 6 Test arrangement for a ship moored by linear

springs

Some examples of the test results are shown in Fig. 7 in comparison with the calculated values. As shown in these figures, fairly good agreement is obtained between the calculated and the measured values. The influence of the vertical spring is remarkable in pitching motion and reduces the amplitude of motion. Under the influence of the hori-zontal spring, the natural rolling period and the rolling amplitude at the synchronous point tend to become smal 1er. In thê wave length range from AIL = 1.5 through 2.5, the amplitude tends to be larger, by the coupling effect of

swaying motion.

4.2 Floating body moored by a spud

As a case of special mooring configuration, model test in waves was conducted for à floating body moored by a

0.5 Heave 0 1.0 2.0 AIL Sway 1.0 o -O -- 1,0 20 _AIL O Case-1 hw/L1/50,p=18O

Fig. 7 Motions of a ship moored by linear springs

L2.500m

0. 622 in d=0.149m 4=232.2kg

GM=0. 124m

Fig. 8 Test arrangement for a floating body moored by a spud

Spud such as in a case of dredger. Principal particulärs of tested model and the scheme of the test aftangément are

shown in Fig. 8. The mooring system is asymmetric with respect to the y-axis; that is, the floating body are moored

by a.spud at fore-end and by linear coil springs at aft-end.

In this case, the lower end of the spud is assumed to be a pin jointed to the bottom of the test basin for simplicity.

The spud restrains surging, swaying, rolling, and pitching of

the floating body, while it keeps the body free to heave and yaw at the mooring point. The stiffnessof the restraint by the spUd is very high comparing with usual inoóxing

configuration such as the mooring chains.

Some examples of the test results are shown in Fig. 9 in comparison with the calcUlated values. The influencé of the spud is especially significant in the coupling of surging and pitching motions. In longer wave length range, there occur resonances in surging and pitching motioñs, and their amplitudes are one order Of magnitude larger than theones

of a usual floating body. Fairly good agreement is obtained between the calculated and the measured values, even though some discrepancy is found near the resonance point

of surging and pitching motions.

4.3 Two floating bodies brought alongside freely The present calculation method can also be applied to freely floating bodies brought alongside within close proxi-mity. In this case, motions are affected offly by the hydro-dynamic interaction between floating bodies. Model tests.

MTB158 July 1983 20 _AIL Case-2 hwILII5O,p90 n 1.0 0.5 0 Pitch 1.0 2.0 AIL 180rn Meas. Cal. Without spring o With spring ln 90

LO 0.5 270 A Ship-i Sway 2.0 - 4.0 AIL 6.0

Fig..9 Motions of a floating body mOored by a spud

Meas. Cal.

CaI.for the independent ship (no iñteraction)

Ship-2 Sway -n 1.0 w 0.5

w

L5 0.5 o

Fig. 11 Motions of ships broughtalòngside freely

Ship-i Ship-2

were carried out, for this alongside condition. The arrange-merit of model test and the principal paiticulars -öl tested models are shoWn in Fig. 10. The two ship models were selected t° be dIfferent in shape and size. The distance Of parallel separation of the models was set to be equal to the breadth of the smaller one of the tested ship models.

Some examples of test results are shown 1h Fig. il in comparison with the calculated values. The calculations

Ship - i Li = 42m B =0.841m d =0.210m 4 =584kg GM= 0.251m Ship -2 L2 = 30m B =0.472m d =O.183rn 4 =220.6kg GM= 0.060m

Fig 10 Test arrangement

for ships brought

alongside freely _{Fig. 12 Test arrangement for connecte} ships by semi-rigid links

Ship models Ship-I L=3.m B = 0.472m d=0.i83m 4 2206kg GM= 0.067n Ship- 2 L = 3.0v, B = 0.493m 0.194m 4 = 237.0kg CM 0.067th Shipi

Fig. 1-3 Connecting link model with a'dal stiffness

were made with and without hydrodynamic interaction

between the two ship models. In shorter wave- length range,

motions of the two ship models are extremely complex because of hydrodynarruc mteraction The influence of hydrodynamic interaction is especially remarkable in mo-tions of smaller ship. When the hydrodynamic interaction is taken into account in the calculatiôn, fairly good agrçé-ment is obtained between the calculated and the measured values.

4.4 Two floating bodies conhected alongside witii semi-rigid link

As a case of cOnnected floating bodies, model test in waves was camed out with two ship models connected by four semi rigid links The prmcipal particulars of ship models and the arrangement of test are shown in Fig 12 As shown in Fig 13 connecting link model has only axial stiffness, of which spring constant is 0.1 kg/mm, and at connecting points on the both ends it is free to turn about

the -points.

Some examples of test results for ship motions and axial displacement of the links are shown in Figs. 14 an 15 in comparison with the calculated values In beam sea condi tion the calculations were made with and without the hydrodynamic interactions between the two ship mädels.

By the influence of connecting link and hydrodynamic mteraction very complex coupled motions of the two ships are caused which differ from the motiotis of an independ ent ship model. T-he influence of connectiòn is- especialW sigmficant in the coupling of swaying and rolling motions In shorter wave length range, the influence of hydrody-namic interaction is observed, but not sO remärkable in

p Meas CaL i35

A --

i80 180 O 135 20 Surge hwIL=1Ii00

ír'\

w-10 c9/;' çrtY A 2.0 4.0 AIL 6.0 w-Heave O 2.0 4.0 AIL 6.O 10 Pitch -IC 5 ci/

,/

\"

\'o

A 270-*! +90V 1.0 _AILi -1.5 05 1.0 _AÌLj Ship-i Ship -2

1.0

0.5

Ship-i

Sway

,a Meas. Cal.

9O O _{* * With hydrn. interaction} 135

--O _i.o 2.0 AIL 3.0 0 1.5 dJ Heave 1.5 v.1.Ò (/I r , o/AA A 0 1.0 2.0 AIL 3.0 6,0 RoIl 4.0 o

/

4.0 0'

_,

2.0 0 A 2.0 1.0 2.0 AIL 3.0 Ship- 2. 1.5 Sway 0.5 0--- 1.0 6.0 Roll 8 hIL=1I50 o

80

?.P AIL 3.0

Fig. 14 Motions of ships connected by semi-rigid links

/

2.0 Ljnk-3 4., 1.0 4.0 2.0 1.0 .5 3.0 0 1 s s o -- LO 2.0 3.0

AIL AIL AIL

Fig. 15 Axial displacements of semi-rigid links

O

experiments as in calcUlations., The effect of interaction by connecting lmk seems to be dominant in companson with that ofhydrodynamic interactiön.

Fairly good agreement is obtained between the calcü-lated and the measured values even when the mfluence of hydrodynamic interaction is neglected in the calculations.

4.5 Floating bodies connected rigidly (Catamaran)

Assuming the stiffness of the connecting member to be very high, the present calcUlation method can be applied to floating bodies -rigidly connected like a catamaran. As an example of the present case model test in waves was carried out for a catamaran(8). The scheme of measuring arrange-ment and the principal part!cuinfs of tested model are shown in Fig. 16. The two hulls were connectôd each other at the midpoint by way of a block gauge. to measure wave loads acting on the cross structure.

Some examples of test results are shown in Fig. 17 in comparison with the calculated values. In beam sea condi-tion, the calculations were carried out with and without the hydrodynarnic interaction. For the rest of the wave incident angles, the calculations were made without the hydrödynamic interaction.

In shorter wave length range of beam sea condition, the effects of hydrodynamic int6raction on motions and wave loads are remarkable In this range, therefore, the

hydro-Sway 3.01- o - I 2.0 LO O -. 1.0 2.0 AIL 3.0 15 Heave í

0__--Single hull L =4.5'm B = 045m d = 0.270m 4 =635.5kg Wave load detector

Fig. 16 Test arrangement for a catamaran

* with hydro. interaction

0 1.0

15 Vert, shear forte

A

0.5

MTB158 July 1983

1._0 2.0 AIL 3.0

Fig. 17 Motions and wave loads acting on cross structure of a catamaran

dynamic interaction is not to be neglected in the prediction of motions. and wave loads of catamaran. in other wave conditiòñs, however, fairly good agreement is obtalned between the calculated and the measured valUes, even though the hydrodynarnic interaction is neglected in the

calculation.

4.6 Several floating bodies connected with pin-joints

As a further example of multi body problems a case of floating bodies connected with pin-joints in series wäs examined. Thi is referred to the case of an articulated oil boom. The scheme of measuring arrangement and the pnncipal particulars of tested model are shown m Fog 18 the tested models consisted of five semi-immersed hori-zontal circular cylinders jòinted at the ends with pins, and shearing forces acting on pin were measured by strain

gauge balance(9).

Some examples of test results for motions, and vertical

and lateral shearing fOrces are shown in Fig. 19 in

compari-son wIth the calculated values. Simplifying assumptiOn is made that the articulated boom in this case is arranged in a straight line. Motions of the boom element in the middle; änd jointing forces between the adjacent boom elements are shown in the same figure. .The vertical 'and the lateral shearing forces are large in oblique sea condition, and tend

AIL 3.0

2.0

1.0 2.0 AIL

5 Lot, bending moment

oo 3.0 Cal. * p Meas. 90 0 135 A 80 o

15 Lot, shear torce

0a

10

0.5 0.5

e135 90

Fig. 18 Test arrangement for articulated booms con-nected with pin-joints

Meas. CaL 90

o

-60 A

--1.0 Pitch O5 0 -- lo A/B 20 Detail f' 200 / '1 " j Universal joint, L = 1.296m B =0.216 rs ii = 0.108m 4=23.8kg t Measred element 0.2 -

/1

Lateral force q

\

ra. I \a

/

A 0 - 10 A/B 20 Vertical force 0.2- 1-' q A 0 10 A/B 20

Fig. 19 Motions and jointing force of articulated boom

to be larger in the range where the mötiòns of the boom element are not so large Fairly good agrement is obtained between the calculated and the measúred valúes.

5. Concluding remarks

In the present paper, results of a series of systematic studies weré described on motions of floating bodies under composite external loads. And â unified analytical method for prediction of the motion was proposed. In o±der to verify the applicability of the method, model expernients were carriéd oùt for several typical cases of moored or connected floating bodies Fairly good agreement was obtained between the calculated and the measured valúes; practical utility ôf the present method, therefore, was confirmed. So far as the composite external loads can be approximated by linear dynamic system, the presentj meth-od appears to have extremely wide variety of applications to all types of "connected", "moored" or âny other com-posite configurations of main body and additional ones including an extreme case of ngid connection as in a cata maran hull. The prediction calculation by use of the present method therefore is fairly useful for fundamental design of various types of floating bodies with external load

sys-tems.

References

[kegami K. and Matsuura M., Study on Motions of a Floating Body under Composite External Loads, Jour. Soc. Naval Arch. Japan, VoL 144 (1978)

Ikegami K. and Matsuura M., Study ôñ MotiOns of Floating Bodies under Composite External Loads, International Sym-posium on Hydrodynamics in Ocean Engineering, Trondhelon (1981)

Salvesen N., Tuck E.O. and Fältinsen O., Ship MotiOns ánd Sea Loads, Trans. SNAME, Vol. 78 (1970)

-Martm H C Introduction to Matrix Method of Structural

Aliâlysis; McGraw-Hill, lñc. (1967)

Ohkusu M., On the Motion of Mültihull Ship in Waves Trans.

West-Japan Soc. Naval Arch., No. 40 (1970)

Ohkusu M., Hydrodynamic Forces on Multiple Cylinders in Waves International Symposium on the Dynamics ofManne Vehicles and Structures in Waves, London (1974)

Ohktisu M., Ship Motions in Vicinity of a Structuré, Proc.

BOSS, VoLl, Norway (1976)

The Shipbúilding Research Association of Japan, Investigation into Catamaran Suitable for Sea Main Transport System, Tech-nical Memorandum, No. 322 (1979)

1

The Shipbuilding Research Association Of Japan, Study on

Performance of Oil Spifi Control Equipments on Sea, Technical Memorandum, No. 55R (1977)

Appendix A Notes on the tansformation of coordinates As shown in Fig. A-1, it is assumed that the vector-of sixdegree-offreedom motions 7) of a rigid body is given in

reference to the coótdinate system o-xyz, and tlat the

coordinate system ö-9 is the parallel translation o o-xyz. And for linear approximation assumption is made that the motions are sufficiently small. The vector of motions 7) which is transformed from o-xyz into ô-llyi, is given as follows by use of the transfòrmation matrix T:

= T,,ij where

I

'IA

L

0I

e

Fig. A-1 Parallel translation of

i coordinate -1=,

0=

I 0 0 0 Q

lt

O -0 0 O O 1 0 0 (À-2) o o o Zo _Yo

A z0

ü x0 Yo 'X0 O

Using the same transformation Ínatrix, the force F acting upon the point ô is transfôrmed into the forôe.

F with

respect to o-xyz as fòllöws.

(A-1)

y Fig. A-2 Rotation

ocoordi-nate

Fig. B-1 Coordinate system for calculation of hydrodynaniic interaction

MTB158 JûIy 1983

section, the incoming waves on the right hand sectioñ and

left hand section can be described as follOws,

A -ikP Rte -4ikP

-

Lr"Rre - -31/cF e R

A(RL)J -R A(LR)J I where

A1 :

'wave amplitude ratio generated by j-th

mode oscillation of the right hand section reflection coefficient of the right hand sectión and left hand sectiôn, respectively

k : vave number

2F : spacing between the centres of sections

The wave amplitude ratio and the reflection coefficient can be calculated by use of Kochm function that can be obtained by solving the radiation problem of each section

Radiation forces induced on each section by incoming wave can be calculated by use of Haskind-Hanaoka's relation, .and

they are just the hydrodynathic interaction force between two sections. In the case that the left hand section is

oscil-latingin the

j

th mode with unit amplitude and right hand

section is restrained, the incoming waves on the right hand section and left hand section can be obtained in the same

way. They are described as follows,

-3ikP e L - -4ikP AR,.,AL,. A(R,L)/ i +

'jk

AL, e. L -41/cF i' - -4ikP where

wave amplitude ratio geñerated by j-th mode oscfflatioñ Of the left hand section

Wheñ the two sections are restrained in waves coming from the rights the same approximation method can be applied. Wave amplitude progressing toward the left and

the right between 'two sections are described as fôllòws,

A(RL)-

R ARt _-41/cF (B5)

I

(B-6)

-41/cF

1-l1L,.IRre

transmission coefficient of the right hand

section.

AppendixC Linearization of mooring and connecting members with strong non-linear property For mooring and connecting members with large non linear property, model test was carried out, and considera-tions were made on the linearization technique.

1. MOdel test

Model test in waves was carried out for a barge moored by several kinds of spnngs with non linear characteristics as

follows:

(1) Case-1 : Moored with linear spring.

(B-l) ('B-4) R (L-R) where ARt -2ikP IIL,.IiRte

F=TtF

(A-3)

Next, it is assumed that the coordinate system ö-92 is a rotated system from o-xyz as shown in Fig. A-2. Defining the transformation matrix T,. for rOtatiOn of coo±dinäte, the relation between the motiOns 1 and , in reference to o xyz and ô j)f respectively is given as follows

(A-4)

where

1-]

(A-S)

For instance, when the rotation is indicated by use of well-known Euler's angle (a, ,'y), the matrix 1' in Eq. (A'5) is

given as follows.

ëosy sin)' O

cosa O sina

cosß sinß O

¡'y

_{sin)' cos'y O}

O i O

sinß cosß O

O O i sina O cosa O O i

(A-6)

The forces F and F have the similar relation to Eq. (A-4) in reference to o-xyz ando-9. respectively. Namely,

F T,.tF

(A-7)

In general, to transform the coordinate system o-xyz into ô which both are the arbitrary coordinate fixed on a same rigid body, the transformation may be conducted in sequential order of parallel translation and rotation. Using the aforementioned transformation matrices T and T,., the transformation matrix of coordinate T as a whole is

expressed as' follows.

TT,.T

(A-8)

Appendix B Hydrodynamic interaction between two sections

We suppose that the right hand section of two sections illustrated in Fig.B-1 iS oscillating in the /-th mode with unit amplitude, while the left hand one is restrained. Ac-cording to Ohkusu s approximation method based on the assumption that there exists a hydrodynamic interaction between two cylinders with respect to the progressing wavCs generated by the oscifiation or the reflection of oñe

Case- 1 Case2, Case-2' Case-3, Case-3' R22 o Case-1. C /Kc

iCase-i.

B 'KA Casé-1,A (E- method) F .,O+y ro 104

J(o)d9 f()dy= Kydy

o so-ni o

/

/

/

/

/

Case-2, B±C R22 / R22 Case-,2,A+C KA+C I KB.I-C

/

/case-2'. A+C _K Case3,

Il

''

Case-3, A+C YO YA o L=2.5rn B=62i.8mrn d = 149.4mm 4 = 232.2kg

Fig. C-1 Mooring conditions in model test YA (F-method) J(Yo+ YA) = j(YoYA)=KYA YO-E YA lows,

rYo+YA (IO _(IyA

Jf(y)

dy

-J

fl,y) dy Ky dy

O _YOYA O

where

y: Swaying mOtion of mooring point

YA Amplitude of swaying motiOn of mooring

point

YO Mean positiOn of swaying motion of

mooring point

K: Equivalent spring constänt f(y): Nôn-Lifiear restoring force

Following equation can be obtained from Eq. (C-l)

JYOYA Yò+Y, tf(y)dy O 2 _IYO+YAfO,)d (C-3) YA2 Jo

Eq. (C-2) iñdicates the relation between, YA añd y0.

Equivalent lineanzed spring constant can be obtained by

Eq. (C-3).

F-method:

This is the method that maximum restoring forces ai é to be equivalent in plus side and minus side of motion.

Equation of the method can be given as follows

f(Yo+YA)f(YoYA)=° (C-4)

Kn±f(y0fy)

(C-5)

YA

L-mèthod:

This is the method that equivalent spring constant is obtained from the approximation of restoring force by thé least squares method.

$YOYA

{fy)_K.y_yo)}2

_{dy+rnin. (C-6)}

YOYA

The principles of stationary can be described as follows.

1Y0+YÂ{fo,)

K-y--y0) dy

O (C-7)

J YOYA

f

YO+Y4y.

fQi) K.(yy0) dy =0

(C-8) J YoYA

Following equation can be obtained from Eqs. (C-7) and (C-8).

fYO+YA dy O

J YoY

Fig. C-2 Linearization method of non-linear restoing force

/ y 010± YA

/

(L- method)

{f() - K(yBo)}dy.min. 110s.l BOBA O / / /

/

f(y)K(YYo) F / (C-l) (C-2) (C-9)

Spring Restoring coefficient

Spring A RA (=Ko) 0,0216kg/mm Spring B KB(=5K0) 0.1080 Spring C Kc( 20K0) 0.4320

Spring Al-C KA+C(21Ko) 0.4536

Spring B+C KB+c(25K0) 0.5400

Case-2, Case-2' ; Moored with symmetrical non-linear property, that is referred to mooring to dol-phin with fender.

Case-3, Case-3' ; Moored with asymmetrical non-linear property, that is referred tO thooring to pier. Arrangement of model test is shown in Fig. C-1. Model test was conducted in beam iaves. Mooring point was fitted on the center of gravity of the model. Therefore, added restoring force due to mooring spring acts in only swaying direction Three kinds of coil spring Spring A

Spring B and Spring C which had different spring constant were prepared. Non-linear chafacteristics were given by their combination, and "Case-2-, B+ C" means mooring condition that Spring B and Spring C are combined in Case 2.

2. Linearization of non-linear restoring force

Three kinds of linearization method were examined for non linear restoring force in companson with test data obtained in abovëmentioned model test. These three

rñeth-ods are shown in Fig. C-2. They are described as follows.

(1) E-method:

This is the method that potential energys by restoring force of mooring are equated in plus side añd minus side of motion. Equation of the method can be given as fol

"C

50 100

Sway ampl8ude l'A (mm)

(a) Case-2 and Code-2'

Fig. C-3 Comparison of eiuivaIent linear spring constant

of non-linear mooring spring

Case-2, A+C --a---0 2 4 Estfmated value Case-2'. A-I-C - Case-2, A+C Case-2', A+C Case-2, 8+C o E-method A F-method O L-method o E-method A F-method OL-method F-method L-method 25 E- method F-method L -method 0-method 20 150 15 10 5 Case-3, B+C - Case-3,A-I-C Case-3, B+Ç Case-3', B-I-C L- method F-method E, L-method method ° E-method, L-method A F-method A-4 6 8 0 2 4

Esthmated value Endmated value

Fig. C-5 Comparison of measuredand estimated values of sway amplitude

K=-3-3- IY0+Y4fd

2YA J YO-YA

Equivalent linear spring constants calculated by abovementioned linearization method for non,linear mooring system used in model test axe shown in Fig. C-3. Equivalent spring constant is a function of swaying amplitude. In Case-2 and Case-2', equivalent spring con-stänt becomes larger as swaying amplitude increases. In 3, it is independent of swaying amplitude. In

Case-3', it becomes smaller as swaying amplitude increases.

In Case-3 and Case-3', the characteristics of mooring spring are asymmetric, and the mean position of swaying motIon is Shifted to softer spring side. Therefore, the mean position of swaying motion Yo are obtained as a

function of swaying amplitude YA as shown in Fig. C-4.

Calculated values of swaying amplitude by use of equivalent spring constant were compared with test results. As an equivalent spring constant was a function of swaying amplitude, it was required to decide it to correspond to measured swaying amplitude in model test

Emethod L- method F-method (C-1 O) - Case-3 .A+C Case-3 . 8-I-C -- Case-3' B+C 50 100 150 Sway ampltude PA (mm)

Fig. C-4 Mean position of swaying motion

E E 540 520 F-method E, L-method F-method E. L-method Case-3, A-f-C method, L- method method

7

MTB158 July 1983 F-method ,L-method 40 60 Eshmated value (mm)

Fig. C-6 Comparison of measured and estimated

mean-position of sway

before motion estimation. Solving Eq. (2.4) by use of that spnng constants estimated values of swaymg ampli-tude were obtained. Comparison of swaying ampliampli-tude between estimated and measured values were made, and shown in Fig. C-5. Also comparison of mean position of swaying motion are shown in Fig. C-6. For swaying amplitude, estimated vaines by use of Fmethod seem to be smaller than measured values, and L-method seems to give a fairly good agreement between estimatéd and measured values. For mean position of swaying motion, a fairly good agreement can be obtained between

esti-0 50 100 150

Sway amplthide YA (mm) (b) Code-3 and Case-3'

4 6 Eimated value 10 150 E - 100

Case-3, A+ 50- 2-o Case-3. B+C A/L-6.0 i i Case-3', ß+C Sway ampbtude 5.0 10'hii=5cm 1 3.5 K0 =0.0216kg/mm 5Ko 10Ko Spring constant

----: Sway amplitude vs. linear

spring constant

---: Linearized constant of non-linear

spring for sway amplitude Case-2, A+C Case-2,A+C 3.0 I Case-2,81-C 15Ko 20Ko (kg/mm)

Fig. C-7 Sway motion and linearized spring cónstant

AI L= 2.0 1.5 1.0 0.8 0.6 '25Ko

mated and measured values by using E-method and L-method. Therefore, it seems that L-method using the least squares method should be applied as linearization method of non-linear restoring förce.

In case where the characteristics of external lcads have very strong non-linear property, estimated value of motion can be obtained by use of linearization method of non-linear property as shown in Fig. C-7. Solid lines indicate relations of linear spring constant and swaying amplitude that. is obtained by solving Eq. (24) for a certain wave length. On the other hand, broken lines indicate relations of equivalent linear spring constant and swaying amplitude that is calculated by use of Eq. (C-10). Therefore, estimated value; swaying amplitude and equivalent linear spring constant for a certain wave

length, can be obtained as cross point of solid and

broken lines. Estimated value of swaying amplitude obtained by ábovementioned method are shown iii Fig. C-8 'in comparison with the measured values. A fairly good agreement can be obtained between estimated and measured values, though a little discrepancy is found in

wave length range near resonance period of swaying.,

5.0 4.0 3.0 2.0 1.0 5.0 4.0 33

6.0 Moored by linear springs (Case-1. Case-1 2.0 1.0 o 1.0

J

Moored by non-linear spnngs (Case-2, Case-2' Measured Computed Case-l.A

O

-Case-1,0 Case-1,C a (hiv 4-6cm)(hiv Scm) A \ --1.0 2.0 3.0 4.0 -- 5.0 AIL r

,

r

r

r

Measured Computed Case-2, A+C

o

-Case-2', A+C A Case-2,81-C O (6w 4 -6cm),( hw= 5cm)

\

O A",, A 3.O - 4.0 5.0 6.0 AIL 6.0

Moored by non-linear springs I ¡

(Case-3. Case-3' ) _/ J 5.0- ' I 30-JA 2.0-

i-/

o O

--

o on a I I I J 2.0 3.0 4.0 5.0 6.0 AI L A o' o'

Fig. C-8 Comparison of sway amplitudes for-various kinds

of mooring springs 250 200 E E 150 100 O 1.0 A 't Measured Computed I Case-3,A+C

O

-Case-3, B+C A 5Case-3'. B+C D

\

(hs=4-6crn (ho'=Scm)