Investigation of seakeeping of catamarans

3 JUU1975

ARCHIEF

Investigation oÍ Seakeeping of Catarriarans

By

Professor Dr. F. Tasai

Draft for

the Scientific Methodological Seminar on Ship HydrodynamIcs in Varna,

Sept-ember, 24, 1974

Lab.

y.

Scheepsbouwkunde

Technische Hogeschool

Delfi

1. Introduction

Recently many types of catamaran-configurated ships, ferry boat, fishing boat, offshore oil drilling platform, oceanographic research vessel and. submarine rescue ship.

1) 2) 3) etc, have been built ' '.

As mentioned by Hadler2), the catamaran is classified into two distintive groups according to hüll shàpe:

a) Conventional catamarans have conventional demihulls symmetrical or asymmetrical about their own fore-aft

vertical centerplanes such as ASF 21

Modified catamarans have hulls of small waterplane area, symmetrical, or asymmetrical. They include special

con-figurationsuch as Littori Industries TRISEC.

The essential advantages of the catamaran are the large

deck area 'and high transverse stability. On the òther hand, there are some demerits in the cataniaran configuratioñ even fpr the seakeeping characteristics in waves. That is,

The decrease of the natural period of roll due to the

increase in the overall beam.

This results in the stiff rolling ànd induces high

lateral accelerations.

The complicated wave loads on the bridging structure

and high possibility of occurtence of slamming impact on the bottom of bridging structure,.

It seems that the structural probleths with respect

to the above dynamic loads delay the development of the ocean-going large conventional catamaran

Accordingly, many seakeeping researches for the new catamaran of the cenceptof the low-waterplane-area con-figuration, which corresponds to the type of (b) group as mentiohed before, are now being strongly performed and

dis-6)

cussed at the Advanced Marine Vehicles Meeting in America.

The investigation of seakeeping of the. catamaan has

been pex formed from several years ago, by following the research technique for the seakeeping of the monohull ship.

In this paper, I will áttempt to review the existing

2. Theory

2.1 Motion

We consider a catamaran advancing with forward speed

V in regular oblique waves.

The Cartesian co-ordinate systems are defined as shown

in Fig.1. In Fig.1, 00-X0Y0Z0 is the coordinate system fixed in space, 01-XYZ the coordinate system moving with ship's speedV and O-XZthe coordinate system fixed in the

ship. _{O-xy is the painted load water plane 'and O is the} point of' intersection of that water plane and the vertical line through G the center of gravity of the ship when it floats in the still water. is the ship's heading angle with the incident wave which advances in the direction -X0.

For the configuration of a section of the catamaran, we use the following nOtations. B the sectional

over-all beam, 2p the distance between the centerlines of the two dernihulls (so-called separation distance)', and B and T are respectively the beam and draft of the demihull.

Assuming that the fluid has irrotational motion we can introduce a velocit potential of regular wave progressing

in the direction of -X0 which is given y

e'° e«'°0

+ wt)

Moreover, the. subsurface of the regular wave is also

given by '

=

e° eo

+ wt)

=

r e°

e.(KX cos X -Ky sin X + ut) (2)

where is amplitude of wave, K W2/g = the wave

number, X the wave length and W the circular frequehcy of

wave.

On the assumption that the ship's motions are small, the equation (2) can be approximately transfórmed into the. followitig equation with reference to the oordinate system fixed in the ship, that is,

=

_e1

_e

K(X00S X - y.sin X)+ wt}

₍₃₎

, where

W

+ KV cos

x = w(l + cos X) (1f)

If we let T1.k(t) for _k 1, 2,

_3,

6 represent the

motion of the ship In surge, sway, heave, roll, pitch and yaw respectively, a general form of the coupled equations of motion in six degrees Of freedom can be represented in

the form of

A

). +B

.

+Cj j} = Fe

ek)

(5)

fOr k = 1, 2, ..

6.

In the equation

(5)

_{is the mass}

or mass moment of inertia of the catamaran. A1 , B and

are respectively added mass, damping coefficIent and

restoring constant. Fe _{is the amplitude of watTe exciting}

5

In the category of linearised theory, we can analyse the ship motions by dividing them into two groups, that is,

symmetric (or longitudinal) and anti-symmetric (or l.ate'al) motions. Surging,, heaving and pitching Iiiòtions belong to

the former, and swaying, rolling and yawing motions belong

to the latter. Each Of the groups is compdsed of three.

coupled linear differential equations and the mutual coupling effects between the two groups are neglected because they are small quantities of the 2nd order.

In the symmetric motibns, however, the surging motion is usually analysed by the equation of motion of one degree of freedom, and heaving and pitching motions are treated

by the coupled equation of motion.

The explicit forms of the equations of motions in ob-lique waves are given as follows:

Surging motion:

(M + A11), +

C11 = Fe1

eet +

i)

₍₆₎

Heaving and pitching motions:

(M + A33)r13 + B33 + C33 fl3 A35 fl5

+ B5

fl5

+ C35 fl5 = Fe3

ewet +

)

(J5 + Ass)rìs + B55 _n + 055 _fl5 A.5 + B5.3 _n

Swaying, roiling and: yawing motions: (M

A22)2 + B22

₁₁₂ + (A2 - M + B2 T1 + A26 n6 + B26 _{116 =} Fe2

et +

2) (J

+ A)n + B

+ _flz+ + (A+2 - M Zq.)112 +

B2 Ti2 +

A6

116 (J6 + A66)6 + B66 6 + B6 114= Fe6

et +

(8)

M is mass of the catamaran; J, J5 and J6 are mass moment

of inertia, about the O , O y and OZ axes respectively;

r

2.

is the . coo'dinate of the cente. of gravity G.

In the equation (6), we usually neglect A11 and calcu-late Fe1 by usIng the Froude-Kriloff's theory. _{In the}

equations (8), fl, fl an 'n6 are defined as the angular

displacements of the catamaran àbout, the origin O.

Based on the strip theory _the

hydrodynamic quantities, such as ard Bk, can be

calcu-lated by making,, use of the two dimënsioial data of long cylinders, and shown in Table 1.

In the Table 1.,

f

_{is the density of the water, .g the}

gravitational acceleration, W the displacement of the cat-amaran,M the metacéntric height, and 9 and are

co-ordinates of the forward end and afterward end of the

cat-amaran. _{Moreover, lowercase letters mean sectional}

hydrody-namic coefficients which are shown in the appendix of the + BLf6 _{Ì6 =} Fe

et

reports of 'eakeeping cbmmitteé in eleventh,, twelfth and thirteenth ITTC. In the evaluation of the sectiolial

hydro-dynamic coefficients, the transverse hydrohydro-dynamic interaction between the twohulls must. be taken into account. The

re-suits of theoretical calculation for the two-dimensional

hydrodynamic. co:efficients are described and dic.ussed in Section 2.2.. _{Ft.irtherinore, the comparison of the theory} and exerirnent Ï.s given in Section 3.. L

The wave-exciting force and moment for a catamaran are calculated by the Salvesen, Tuck and Faltinse.n's strip

theory)

_{developed for the monohtìll ship. The computing}

method according to the above theory is given.by Pien and

Leeh1,

_{and NordenstrØm, Fa1tinsn and Pederser12).}

7)9)

When we use the usual strip method ' . for 'computing the wave-exciting force and moment, diffraction _pressures for the longitudinal motions are'calculated by using _the

orbital velocity and acceleration of incident _{wave at the}

mean draft G T on the center line of catamaran hull

(G =

coefficient of the sectional area below the water,, line; T = draft of cross section), and lateral motions àt the draft

of T12, on the center line. _{The. usual strip method' does}

not give good approximate value of _{wave-exciting force}

ex-cept.ing'the longitudinal wave condition''.

It is better for the case of catamaran that _the di.f-fraction pressure should be calculated by using _{the orbital} velocity and acceleration'of the incident _{wave at the mean}

draft of the center line of the each demihull. .

Therefore, when we apply the usual strip _{method, it is} necessary to calculate the two-dmensiona1 radiation _forces acting on the eachcylinder of the _{two cylinders oscillating}

independently, _{in prescribed thodes 6f motion with given} amplitudes and phases. _{Using the results of this radiation}

problem and the orbital velocities with _{phase difference}

of .2 K. P sin X on the mean draft o.f the _{sections of}

demi-hulls, we can calculate the diffraction _{force and moment}

acting on the. section by the same Tnethod as is usually used in the strip methöd

Under the assumption of a linear _{response, the solutions} fór the motions of a ôatamaran _{can be expressed by}

n(t)

k cos (wet + (9)

, where and ä _{are' amplitude and phas angle of the} motion of _{- mode and are obtained by solving the equations} of motions (6), (7) and (8).

2.2 _{Two-dimensional hydrodynamic force and moment}

Two-dimensional potential problem associated with heaving oscillation of a twin-cylinder _{consisting of two}

rigidly connected semicircular cylinders _{was first}

investi-1i+)

_i's)

gated by Ohkusu _{and Wang and Wahab} . They used the

method of multiple expansion to determine _{the unknown}

ve-locity potential.' 'This méthod was first introduced _'by Urseiil6) _{and extended by}

_Tasai17)18)

and Porter19) to

20).

the Lewis form _{cylinder and to the case of arbitrary} cross sections with n-parameter, using conformal mapping.

With the assumption of small oscillation, _{they obtained} the' exact calculation results of the _{hydrodynamic forces}

acting on the above two _{cylinders, and have clarified} that there exist following two characteristics Owing to

the hydrodynarriic interaction between two cylinders:

The're are mumerous waveless frequencies at which the

diverging wave generated by the _{oscillation of}

twin-cylinder _{becomes to zero, and therefore at these}

frequencies the wave-making dampiPg _{forces and, also} wave-exciting forces acting _{on the twin-cylinder in}

regular incident waves are zero. . '

There exists the range of frequency _{inwhic1- the added}

mass become.s negative and its _{magitude is large.}

Next, following to the _{same procedure,} Ohkusu2l)

veloped the calculation method of the hydrodynamic forces

on the twin-cylinder consisting of two Lewis form cylinders

performing heaving, swaying ör roiling oscillation in a free

22)

surface. de Jong also dscussed solutions wthout pro-viding numerical results for heaving, swaying, or rolling twin cylinders of symmetric cross section, using conformal

mapping.

On the other hand, NordenstrØm, Faltinsen and. Pe,dersenl2)

23) 21+) 25)

Lee, Jones and Bedel , Takezawa et al and Maeda used, in solving the. two dimensional potentfal problem,. the method of source. dist±'Ibution on the cross section contours of both cylinders. This method is the same one

as was applied for an oscillating single cylinder by Frank.

21+) ₂₅₎

Takezawa et al and Maeda calculated not only

the radiation forces but also exciting fôrces acting on twin-cylinder in beam waves by using the Bessho's extended

26) 27) 28)

formula of Haskind-Newman's relation (Appendix B.).

2.9)

Ohkusu . developed an approximate calculation method based on the assumption of an interactiOn between the two

cylinders existing only with respect to the progressing

waves generated b the oscillation or reflection of one cylinder. _{This effective method is in detail described in} Appendix A. .

Using the source distribution _{method, Kim3° calculated} the radiation forces acting _{on the each cylinder of the two} cylinders oscillating independently _{eàch other, in prescribed}

(arbitrary) modes of motion with given amplitudes and phases on or below the free surface.

Kim's method and Ohkusu's approximate fliethod are very

convenient to evaluate the motions of the individual bodies

when thei are elastically coupled or unconnected.

Shown Figs.2 _{6are the results of added mass and}

damping coefficient etc. for twin-circular-cylinder of

radi-us a heaving and swaying in a free surface calculated by 1L+)

Ohkusu

As seen from the numerical calculation results shown

in Figs.2 6, the added mass and amplitude of diverging

wave are markedly peaked at. the values of KQ..in the vicinity

of the characteristic frequencies given by the following equations(lO) and (11).

(for example, _{1.6 forj the case of 2P/,= 6.0 in Fig.2).}

As the oscillation passes across the characteristic .frequen-cy.the added mass becomes negative ma _{very narrow band} of frequencies and the amplitude of diverging _{wave and}

ac-cordingly. the damping coefficient falls down to zero at a certain frequency.

At these characteristic frequencies the motion of the

fluid between the two cylinders is strongly _{excited by the} oscillation and these cönditions corespond _{to the natural}

mode.s of the motion of the fluid between the two vertical

walls of 2(p -a)(or 2p _{- B) apart, and these characteristic} frequences can be obtained fròm the following _equations: io), il)

Heaving

2K(p -0)

2nit.

circular section

KT

2nTt

-.

KT=

n'lt

-

3T)

, where (h

1,

2 12

arbitrary symnietric

cross section

Swaying and Rolling

2 K(p -a.. )

:

circular section

arbitrary symmetric

(1.1)

cross section

and B and T are breadth and draft of dernihul].

On the other hand, from theAppendix

-of twin-circular_cylinder.

_{, putting}

A A90,

the case

= A:,

E31

= E

= E30, _E

= E= E

_{and PL}

_P

= P, and using

the relations (A-20) and (A-32)

_{we can obtain the wave}

diVerging from the twin-cylinder

_{at Y}

_{in the}

follow-ing forms, that Is, for

the heaving

2(1

et - KY)

(12)

2+e

+e

and for the swaying

2(1 - e)

e

et - KYR)

₍₁₃₎

2 -

(e

+ e

) Wh e r e

S = 2E

- 2KP,

= 2E

- 2KP.

We can know the values, of Ko..wheret'he amplitude of waves

diverging at infinity from the oscillating twin-cylinder

.(10)

become zero and they are given by

= 2c- 2K? = (2n + l)it

(Heaving), ₍₁₅₎

H 2C _{- 2KP = 2n7c (Swaying and Rolling). (16)}

(n = ,

+l,±2,3)

The values of KP which satisfy the equations (15) and

(16) are the waveless frequencies in the vicinity of the characteristic frequencies discussed before.

In the case of heaving motion, putting _{n = O in the} equation(l5) we obtain the waveless frequency which

cör-responds to the zero' s mode of resonant motion of the f1ud (for example, K 0.8 for the case 2P/&

_{= 3}

in Fig.2).

At this zero's mode, the fluid displacement between

the cylinders is approximately uniform and 1800 _{out of phase} with that immediately outside the cylinders.

It is important to note that in the neighbourhood of the waveless frequency at zero's mode, as shown in Figs.2

and 3, the peak values of diverging wave and negative _added

mass occupy a rather wide range of frequencies, as compared

with those which occur at other modes.

Ohkusu's approximate solutions are also given by dotted

lines in these figures.- _{As can be seen from these figures} the results by this approximate method agree almost completely with the exact solution at least for the- range of which the ratio of the separate distance. 2p to radius _{is larger than}

con-venient to apply for the case of multiple cylinders _havihg three or more element cylinders with arbitrary _{cross sections}

as shown in the Appendix - A.

15

2.3 Dynamic structural idading in waves and

hydrodynamic impact

The types of dynamic loads exerted _{on the catämaran}

structure induced by the oscillatory motions _{of a ship and} the fluid surrounding the ships _{are in detail described by} NordenstrØrn et. al'2 _{and Pien and Lee" etô.}

The important iave loads for the hull _{design of the} catamaran are the transverse force V2, vertical _{shear force}

V3,, vertical bending moment Vk and torsional moment about

' axis 'V5 . ,

These force and moment _{Vf can be expressed as f011òws:}

'V1

= + +. F1 + Fej ( 17)

where I _{are inertia forces or moments, C1 the hydrostatic} ones, F _{the hydrodynamic ones and Fej the wave exóiting}

ones. _{For the calculation of the we must obtain the} above force components acting _{on the two demihulls of the}

catamaran respectively.

In the following, we describe the _{method of calculating}

dynamic loads at the midpoInt of _{cross deck of the}

catama-ran oscil44ng in beam waves, and discuss the. dyñamic loads acting on the h'ill of the catamaran advancing in head waves.

2.3.1 Dynamic loads in beam waves

the lóads in e.am waves are shown in Fig.

_7.

The vertical bending uïoment Vk is the monent which

tendsto roJi the hulls about the point Q relative each

other or,' equivalently, to sag or hog the. cross beam.

Positive bending moment is difined as the moment which tends to roll the right hull in a clockwise directiOn or

the left hull in a counterclockjse direction.

The vertical shear force V3 and transverse force V2 are the foröes which respectively tend to heave and sway the hulls relatively to each other. Positive válue of V2

is defined as the force which tends to sway the right hull to the right or the left hl.l to the left. _Positive verti-cal shear is defined as the f rce which tends to heave the. right hull downward or the left bull upward.

For the symmetric twin cylinders, we use the following

notations:

2. _{AO2:. n0, A92= no, .g A02T2}

2 2

w3

pS

=m, PS

=m,

_PS

T2 (18)

P T S M =M0, nc2 T N =

w3

From the Appendis A, the radiation forces and _moments

.cting on the right cylinder R are given by the following equations:

R..

F22 - A

et2

_2R2

m(-2)

₊

(-2),

16

R F2 =

-a4ß (

R.

F2

R F23

_r

A2 T

et+

-, P A,

eE3) ag

n2

'o

tC2 ,

Pg -o

r

A2 R. pg

F3---A3e

3

F-

g (Ä T a41 n o e L

e2

_{+ M0(-fl) + N0(),}

'7 P

e")

.' o

'o

T){A

et

A

e}{E0(KT)

_Ei(KT)}

+ M0(-n2) + N0(-n2 _, F33 A etE3 _3R 3

+ m(-3) +

_{-n ),}

F3 =

_g (A T P

_{e3) a3Rfl3}

+ P{m(-fl3) +

₍₂₀₎

+ rn(-) + n

+ P{th(-n) +

₍₂₁₎

Where

e2{Eo(KT)

_{- E1 (KT.)}} -02L' a3 A

eC3{Eo(KT)

_{+ E1(KT)} =} E1(KT) =

¿ H'(KT) e4'/(j

E0(KT) =

eL<7/(i

), and = -

_{H+(K)2 e4'.}

(22)

cylinders L and R indued by the _{incident wave}

e'

are

immediately obtained from the _{equations (A1I6) (A_L8).}

Thus the sectional dynamic _{1oad at the midpoint of} the cross beam in beam waves are given by the following

formula: a) Transverse force V2 R. L = LF23 + (Fe2 .-- Fe2) A2 fl3 + _{± B}

eWt

+ KF) where A2

2g

e

_{e. {E0(KT)}

_{+ E1(KT).}}

Next, the waVe exciting forces _{and moments on the}

and

and

, where

.0

A23 _{e2, {.E0(KT)}

-Ak3 =PP(Sw2

g ) g A _{(P - T)} =

g BP

- F2

= A2.3 _fl2a. SO x (A etE3 O .'F (m3 w2 -i _Pg -0 3

-r

C A3 e _{{i + E3(KT) E1(KT)}} (2h)

(wt+5

). e + A3. fl e O w) 18 ¿ CO e L){Eo(KT.) _{- E1(KT)}} R i R. L '- F3 (Fe3 - Fe3)

A e2 {i

_{- E2(KT) - É3(KT)}. (23)}

e) Vertical bending moment about

_{Q, V:}

where

=P{p S

n

± p gB rh}

-

(F13 -

F3.h0)

(Fe

- Fe) +

(Fe

- F2)

¿(oit 1- _{S3) (wt + K?)} rj30 e _e

= Pf{(S

w2 _{- g B) + m w2} - _w e' _{{E0(KT) +E1(KT)} x}

o

Lc _o

_.ic°

x (f

A3 e _{T Ak e} k _{+ h0 A2}

_e'2)

Bk = A _{{l +E2(KT) + E3(KT)}} -o .E _O i.c

+ (h0A2

2 T Ak e

k){1

-

_{E2(KT),- E3(KT)}] (25)} It is seen from the equàtions _{(23), (24) änd (25) that}

vertical bending moment and _{transverse force. are afected} only by heaving motion, and the _{vextical shear force is} afected only by swaying and _{rolling motions.}

2.3.2 Dynamic loads _{xerted upon the catamaran} advancing in head waves

The wave-induced _{shearing force and bending moment} an arbitrary section of the catamaran _{hull can be easily}

calculated by the same procedure äs is used for the monohull ship.

ccording to the results of _{computation made by Ohkusu} there occured a large maximum _{value for the longitudinal} bending moment at midship of _{the catamaran of which the} ratio of sepàration distance _{to draft ?P/T was 3.0.}

19

31

Furthermore, that maximum value _{was more than twice} of thè value of maximum bending _{moment computed without}

the hydrodynamic interaction effect _{between two demihulls.} From the equation (18) and (20), _{the two-dimnsionai}

value F3 cañ be expreed as follows:

R F23 -

--A2e

-o

a

c.

((V

J we2 V2

v{(z;n3)

-

V o o o. am an - m23.fl3 - {n23 V + V o V an23 w o - w a +

w(

23 t' flO

+J(m23)5

+ {X.fl23 -

_Vm23

_{V (C 1fl23)}

-

w2.

m3]x

exP.{K.a T

_{+ 1(K}

_{w t)}

_LX

(29)

Next, the two-dimensional coupi±ng moment F3 can be

expressed as follows:

R _..

o

= T-m3(-fl3) Tnz43(-fl3).

(30)

The vertical bending moment V _{about the midpoint Q}

of the cross deck at- a gt-ationcx _{is given .by}

S(fl3 -

fl51_ P

g (n -

n)}

+ Jjdx(31)

20 m23(-fl3) ₊

fl23(fl3).

(27)

Using the equation (27) and thern New Strip Method, transverse force Va is. given by

Putting

m(X) = (T.m3

- h0

rn3)

and

m9

m2s = (w)]2 S(w)dw _{= variance of relative} vertical displacement, 2(w)] S(w)dw = variance of relative vertiòal velocity

and fec is the effectivé height of _{the cross-deck bottom}

from the mean water level. _{In the eqüation (33), S(w) is} the wave spectrum, _{the ratio of the. ela.tive vertical} displacement of the deck to tIe _{wave amplitude. The}

ef-21

(32)

(33)

=

_{.n3 - ho n3).}

J _{can be obtained by substituting m(x) for}

_{m3 and}

n(x) for n3 respectively _{in the êquation((29).}

One type of important load for _{a catamaran is slamming} impact when the bottom of thecross-deck _{hits the water.}

32)

According to Ochi _{, we can obtain the probability P of}

occurrence of slamming undr the condition _{that the relative}

vertical velocity Zr qf the _{forward cross-deck at a station} X to the wave surface exceeds a critical value

₁₄

_(so-called

threshold velocity) when the deck _{bottom reenters the water,}

and it is given by

= exp[- fec2 / 2mos + 2m ].

fectiVe height fec i:s given by

fec. = fc _{- hs - h (3L)}

, where fc is the geometrical height of. the cross-deck

bottom from the still water level, h5 the _{relative elevation} of the water surface due to trim, sinkage _{and bow wave, and}

the dynamic swell upof watersurface _{generated by ship}

motion.

Therefore the number of slamming _{per an hour is given} by 1800 N3 -. P5fm2s / trios. 33) As seen from the Wahab's results of _experiments

for ASR catamaran, decreasing huilseparation _{and increasing}

speed oth decrease te effective freeboard. _{Thi.s fact}

indicates that the statical swell _{up hs increases with the}

increaseof speed and decrease of _{separation. Therefore,}

to avoid slamming, it might be desirable _{not only to raise}

the cross-deck as highas posible but _{also to develop the}

hull form which generates small statical. _{swell up.}

(3.5)

3.

Comparison of Theory and Model Experiment 3.1 Two-dimensional radiation forces

In order to verify the theOry, model _{experiments were}

1L) ₁₅₎

conducted by Chkusu , Wang and Wahab _, Lee, Jones and

23) 2k)

Bedel and Takezawa et al _{by means of the forced}

heav-ing method in the condition where the fluid motIon is expect-ed to be two dimensional.

1k) 15)

Ohkusu , and Wang and Wahab used a catamaran model

consisting of two émic1rcu'1ar cylinders. Lee et a123) used 'four models of'which demihulls are semicircular, rectangular,

isosceles triangular and right triangular cylinders. Takezawa

et al2 _{used two models, the oneconsisting of twoLewis form} cylinders and the other of two waveless form _cylinders.

Qhkusu measured the height of the diverging _{wave generated}

by the heaving of the model. _{The comparison of the calculated}

- ¡S

and rieasured A arc shown in Fig.8.

Wang and Wahab, Lee et al and Takezawa et al measured the forces acting on the model. _{To obtain the added mass and} damping coeff±cient, the measured fOices _{were reduced to}

components which were in phase and _{90 deg. out öf phase with}

the he.aving displacement.

The results obtained 'by Wang and Wahàb' are shown in

Fig.

_9.

'

As seen from these investigations _{we may conclude that}

the results obtained by model experiments and _{those calculated} using the linearized potential theory _{'agree very well.}

3.2 Three-dimensional radiation forôe and wvé-exciting

force and moment

11)

Pien and Lee compared the experimental results of added mass and, damping coefficiênt of _{a conventional}

cat-amaran model with those calculated by the equations shown

in Table 1, and it was fouhd that _{agreement between the}

theoretical and the experimental results is good for the

zero-speed case, whereas some discrepancies _{can- be observed} for the câse of high speed.

Shown in Fig.lO and Figll are heave added mass and damping, and: pitch added mass Pioment of inertia

anddamping

for the ASR* _{catamaran In case of Fn = 0.126. Ít can be} seen that çhe agreement is good with the exception of the lower predicted minimum seen in the damping.

The nondimentional heave exciting force and _pitch

excit-ing moment for the ASR qatamaran rriodel are shown n Fig. 12. The agreement in the heave exciting force is _{quite good,}

however, _{the theoretically predicted pitch exciting}

moment grossly overpredicts the expêrimental results.

*

catamaran intended to service the deep sea rescue vehicle

3'. 3 Motions

The model experiments for catamaran models in wave.s

31) 33)

were performed by Ohkd.su'and Takaki , Wahab et al

3L.) 11) ₃₅₎

Takaki ét al , Pien and Lee and Lee et al

Ohkusu and Takaki3l) _{used the two catamaran models of} TW i and TW 2. _{These have the same demili shown in Fig.}

13

but different separation ratio of

_{2T (2p = separation}

distance, T = draft). _{The value of 2/T for the former is} of 3.0 and of 5.0 fo±' the latter.

Moreover, the experimental results _{were compared with} the theoretical results. _{Theoretical calculation of motions} were carried out by the Ordinary Strip Method. _However, in beam, waves, both radiation force _{and wave-exciting force}

on an arbitrary section were exactly computed _{by the. effec'tive}

calculation method shown in the Appendix - A.

shown in Figs.1LI, _{15, 16 and 17 are the theoretical} and _{.xperiment'a1 results of amplitudes of heaving and}

pitch-ing motions at the Froude number Fn _{O and 0.1. In these}

figures the amplitudes of heave _{and pitch' Z,® are} non-dimensi.onalized by the wave amplitude _{q. and maximum wave} slope respectively and À, L are the wave length and

ship length.

As can be seen from these, figures, there is

compara-tively large difference between the motion amplitudes of

catamarans with and without taking the interaction effect

into account if Fn 1s nQt zero, and it can be concluded

26

that theoretical results are in good agreement with the

experimental results.

The amplitudes of leaving,swaying and rolling motions of TW i and TW 2 in beam waves were measured _{at zeró speed,} and the experimental resutis compared with _{the theoretical}

ones. These results are shown in Figs.18, 19 and 20.

The models drift due to the drifting force _{induced by} waves and accordingly the abscissa of these figures is.we2T/ where We is the circular frequency of _{enòounter of incident}

waves.

The theoretical and measured values of _{motion amplitudes} are generally in good agreement exáeption of _{peak values at}

resonance.

In the neighbourhood of resonant frequency. of heaving

motion, the experimental results _{are much smaller than the}

theoretical results. _{At the experiments of these frequencies}

it was found that the waves between _{two hulls began to}

progress in the lengthwise direòtion öf the model _{and measured} amplitude was npt stationary but varied with _{a time like a}

beat.

In Fig.18, theory predicts that there _{aì'e. three} Ieso-nance points for heaving motion of TW i in the range of

frequency shown in the figure. _{One of these three points is}

the frequency where the _{amplitude of the heaving motion}

of TW i becomes the largést _{as shown in Fig.l8, and the}

other two points are we2T/g

27

two points, the amplitude does nót show _{a peak value because}

the damping force is very large.

In case of the rolling motion, the theoretical _{and the} measured values are also in comparatively _{good agreement in.}

spite of the neglect of viscous damping. _{The above is due}

to large wave-making damping moment for _{rolling motion.}

Shown in Figs.2.l and 22 are heaving and rolling

ampli-tudes of TW i with and without interaction, _{where approximate}

denotes the values without the interaction. _{As seen in}

Fig.2l, if the interaction is not taken _{into account, the}

amplitude of the heaving InQtion

Is

_{not only smaller than}

the exact one, but also the amplitude _{is rather large where} the exact calculations show the _{zero. amplitude.}

In case of rolling motion (Fig.22), _{both the exact and}

approximate calculations give almost the _{same resonant}

frequency and the wave-exciting momênt, _{but the damping}

moment without the interaction is iager and the

approxi-mate amplitude is much smaller than the _{exact one.}

As seen from Figs.2l and22, the _{motions of the} cata-maran are sometimes different from those of _ordinary mono-hull and the motion characteristics _{of the catamaran in} waves can not be acculately predicted 1f the _{hydx'odynamic} interaction effect between two

hulls Is

_neglected.

Pien and Leehl) _{performed the model experiments for}

the catamaran model of Low-Water-Plane _{configuration} ad-vancing with high speed in _{head waves, and compared the} experimental results of heaving and pitching _amplitudes

with the theoretical calculations (Fig.23).

There is a large discrepancy bétween the _experimental

and theoretical results.

The strip approximation may exaggerate the. effect of the hydrodyriamic interaction between two hulls.

11)

Pien and Lee _{introduced the additional damping force} depending on a parameter of WV/g. _{That is, the modified} sectional damping is given by

b33(X)

= b3

(X)

₊ p w S(C)

g (36)

where b33(x) is the original heave _{amping at a cross section} ofX, P the density of water, S(x) _{the sectional area and}

a = 3.0

for the tested model. _{The dotted curves in Fig.23}

are obtained with the above modified damping _coefficient.

3.Li _{Dynamic loads}

36)

Curphey and Lee _{compared the theoretical results}

of dynamic loads for the ASR catamaran _{in beam regular waves} 33)

with the experimental data obtained _{by Wahab et al}

Fig.214 indicates the, prdicted _{and experimental amplitudes}

of the vertical bending moment and _{vertical shear force at} the midpoint of the cross beam as a function of _{the ratio}

of the wave length to overall beam X/B.,,.

It is seen from Fig.211-a and Fig.24-b that theoretical and experimental loading results _{were in relätively good}

agreement for both shape and _magnitude.

4

Correlation between Calculation and Full Scale Measurement

Taking an oppotunity of construction of newly developed iron ore loading station with submersible, column stabilized, catamaran hulls, a systematic study was made on the motion

and strength of the structure in light draft condition

when being towed in the oOean13'3 The loading station "PRIYADARSHINI", which is the object of the study (herein-after referred to as L.S.), was designed to improve the loading and unloading operations offshore of Goa, India.

It is grounded on the sea bed off the port of Goa in approximately 15 meter water depth. Iron ore is unloaded from several barges, accumulated on L.S., and loaded into a 60,000 DWT class ore carrier continuously. _{During the}

monsoon season, the bperation are ceased, and L.S. is floated

and towed into the port for evacuation.

The principal dimensions are shown in Table 2

LL1 _{Theoretical Calculations of Notions and.}

Longitidinal Bending Moment.

The calculation methodrnade on this catamaran in

shallow ballasted draft is the same as that used for calcu-lating motions of ordinary ships. Thàt is, responses of

periodic motions and longitudinal wave bending moment of the catamaran thàt advances With a constant speed V in

29 U

oblique regular waves are calculated by the New Strip

38) .

Method developed by Tasai and Takagi

Two-dithensional radiation forces acting on the unit length of a catamaran hull were calculated by Ohkusuts

method _{The diffraction pressure was calculated} approxi-.mately in the same manner as. in an ordinary ship. That is,

for heave, and pitch motions, diffraction. pressure was

càlcuiated by using orbital velocity and acceleration of

incident wave a.t the depth of T on the centerline of the

catamaran hull, and for sway, roll and yaw motions, at the

depth of T/2 on the centèrline..

Solving these two sets of coupled equations of motions,

the amplitudes and phase lags for pitch, heave, sway, roll

and yaw motions in oblique waves were obtained. Using the responses of the pit.ch and heave, and the value of weight

distribution the response of the longitudinal wave bending moment at midship section of the catamaran was computed by the sáme procedure a was used for ordinary ship.

Next, the water-tank te,sts for the vrification of the calculation were conducted, using a 1/socale model,

in the towing tank (L

x B. x D x

d = 80m x 8m x _3.5m 3m) of the Experimental Station of the Research Institute for

Applied Mechanics, Kyushu University.

The draft of the model was selected at a value equivalent

to that of L.S. in actual towded condition.

- In head waves, heaving and pitcmotions were measured

'i

for Froude number Fn=0.05 and 0.1, and in beam waves,

heaving, rolling and swaying motions were measured

_uner

the condition of Fn=0.

Ari example of the comparison between theory and

ex-pe'iment are illustrated in Figs.2and26. _{In Fig.26, 2p}

is the separation distance.

As there were some differences between calculation

and experiment the dorrection coefficient were introduced.

Referring to Fig.5, the correction coefficients for

amplitudes of heaving and pitching motions of L.S. advanc-ing in head waves with 5KT (Fn=0.08), _{are determined as}

0. 33 for high frequency range in which these amplitudes

extremum in head waves, and for other frequencies, _no

cor-rection is made CFig..7). With respect to the amplitude of

roll, as seen from Fig.2, the experimental value of the roll angle at resonant frequency is approximately 60% of the exact calculation obtained by using the wave-exciting force and moment calculated by the Bessho's fomula given

in Appendix A and B.. Furthermore, the calculated values by the strip method were larger than the exacts ones, and. the latter is about 50 percent of the former at resonant

frequen-cy. _{From these comparisons it results that K, the correction}

coefficient of roll, becomes 0.3 at resonant frequencies.

As seen in Fig.2, the experimental results of swaying amplitudes in beam waves showed fairly good coincidence

with that of exact solution. _{However., there are considerable}

differenciesin value between the exact solution and that

calculated by th.e strip method, and therefore, the authors

adopted the correction method that the calculated value of

swaying amplitude at high frequency range is multiplied by

0.5.

As for the bending moment, based on the results shown

in Fig.27 , KM the correction coefficient was determined.

These correction coefficierts are shown in Fig.27.

14.2 _{Strength Calculation Method.}

On the calculation of longitudinal strength,the

finite-element method using the large size finite-elements was applied

for both cases of subjected structure being assumed to be Rahmen structure and shell structure. These calculated

results of the stress were compared wIth those measured

at the time of launching.

From these investigations, it was found that over-all

strength can be calculated wi.th fairly good accuracy .even if large-size elements are used as a shell structure.

.3 Field Experiment of Loading Station 14.3.1 Outline of experiments in the ocean

The comprehensive measurements of L.S. were conducted

during the period of Dec.28-30, 1971., from the offshbre of Sikôku Islands to the offshore of southeast Kyushu Island. Measurement of waves was also conducthd, simultaneously.

Fig.28 indicate the general scheme of the measuring

system. After the observation boat reached the area of

experiment approximately two hours ahead of the L.S.', wave measurement was conducted by the clover-leaf buoy (see Fig.

29) for about 30 minutes. When the L.S. was towed to that

area,.the measuring instruments on L.S. were started by

remote control from the observation boat. After the wave

measurements were ended and the wave buoy-taken on boad, the observation boat sailed parallel to L.S., änd the angle of

encounter with the dominant waves and towing speed of L.S. were measured.

The measurements of motion and tress of L.S. were continued for about 20 minutes, and the measured data were recorded on magnetic tapes, which stopped automatically at a designated time.

Thus, in some cases there were differences between measured data of L.S. and wave data, 30 to 90 minutes in

time, or a maximum of 6km in distance. However, these differences did not affect the analysis much because no

rapid change of the weather was found during each observation.

14.3.2 Measurement and analysis of waves

The measurements o ocean waves were conducted by using the clover-leaf buoy newy developed by the Research Inst i-tute for Applied Mechanics, Kyushu University!. The buoy (see Fig.29 ) is essentially the same type as that of the

311

National Intitute of Oceanography, which can xneasùre the vertical aöceleratior, slope, and curvature of the wave

surfce.

:.

The measured data are used for estimating the di-rectional wave spectrum.

E(f, ) =

(36)

, where denotes the propagation direction of the component

wave relatiVe to the domjnait wave direction, and (f) denotes

one-dimensional spectrum difined by (f)

=J

(f, ,.

₍₃₇₎

and G(f., Y denotes the diréction distribution furction

that indicates directional energy distribution of waves.

The collected wave data Were digitized by using an A-to-D cnverter arid further processed on the FACOM

_270-20

electronic computer system.

An example of a one-dimensional wave spectrum is shoWn in Fig..30. For estimating the spectra of motion and

lon-gitudinal bending moment from the wave spectra, the measured values of one-dimensional wavé spectra were used directly, but the directional distribution function was tentatively

assumed as

GC) =

_00s2

for

(38)

414. _{Spectral Analysis of Motion and Stress of L..S.}

LI.LL1 Spectral analysts

Th measured data in analog form were digitized at

200 Hz or 250 Hz with A-to-D converter, thên power spectra were calculated using a -stndard program baed on a fast

Fourier transform procedure.. The conditions relating to

the spectral analysts were almost the same as those for

wave data.

14 4.2 Response functions ofinotion and longitudinal

berdi.ng stress in Oblique waves

Assuming that the frequency characteristics of the correction coefficients in oblique waves are the same as

those,given in Fig.27, the response functions of motion

and longitudinal bending stress in oblique waves were

calcu-lated by the strip method and partially corrected by using the results shown in Fg.27.

14.14.3Conersion of power spectra

Within a framework of linear theory, the ship response

spectrum can be given .by

S(f)

= J

ECf

%-Zhc)[ACf,X)]2 dZ,

_r

(39)

where

ECf, ) = directional waVe pectrum.

angle of encounter in which the ship hull meets wi.th the dominant waves

AU',

x)

,rrequency response fnct1.on o.f ship hull 3(f) = power spectral density of ship hull

respoì'se

f _{= frequency :Ln Hz}

(Coordinate systems are shown in Fig. 31

To compare thi.s power spectral density SU') estimated from the wave spectrum with the ofles computed directly from

measured data, it must be converted to the spectrum S'(fe)

as a function of the 'requency of encounter fe. Because

waves were measured a a fixed point, and the motibn nd

stress of L.S. were measured in towing conditions in the

sea, the following conversion,of the spectrum is needed:

S'(fe) =

SU') / (l+LV.cosz )

(0)

the case of oblique following waves (X > 90P),

however, there occurs a difficult problem where, at some

frequencies, power spectral density becomes infinite. To

cope with this, the following approçimation was made for

converting the ship response spectra;

S'(fe) = 3(f) / (1

ff

y

cos

Xpc).

With this assumption the difficulty in he calculation was eliminated.

For the case, X =

lO0

_{and V = knots, for example,} the convertcd spectrum becomes infinity at the frequency

f = O»495

Hz,

fe =

O.248

Hz.

However, this was not so serious, because most of the power density of the ship hull response obtained for the field measurements was in arahge of

_0.23

HZ orbelow.

The values of observed on the observation boat

sailed parallel to the L.S., were used in the äbove

calcu-lations.

14.5 _{Correlation between Theory and Full Scale Measurèment}

The results of theoretical spectral analysis of motion and stress of L.S. vere compared with those 0V the spectral

analysis of measured data. An example of this comparison

is shown In Fig.30 and

_32.

As can be seen from these figures, the calculated values of the mode frequencies of the spectra of ship hull

responses, excluding the one of longitudinal bending moment,

showed good cöincidence with those of measured responses.

An example of thecomparison between the calculated and measured significant values of double amplitudes of the responses of L.S., is shown .±n Table

_3.

It is found that the theoretical values of motion of

L.S. are in good agreement With the measùred values.

How-ever, as for the longitudinal bending stress, there is a

small difference between calculation and experiment.

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