A calculation on the heaving characteristics of a catamaran

ARCHIEF

Lab. y.

Scheepsbouwkunde

Technische Hogeschool

DeIfL '

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'he ch:aoteristics cf the heaving aotion of a catamaran ery coaplicated. compared with that of a sinlc float ship,

:d the theoretical treatment is difficult. _{Only it can be}

ti

true that in ie of th

caaaaran even the rollins motion

:onstructerl with tno neaving motion of each float. _{o if} the heaving motion of a catamaran is known it can be a step to know general motions of a catamaren. In this paper an

approx-te method of calculation on the heaving iot1oì is described.

rnrfl;' T'PTT',1T' _T' rntT' nT'-rn T('

T'TTT '1T'

r--.. .i 4- i. a . . J, . £ .-. . _. ., - '1 . .j

j

.

'or the oimnlicity two-dimensional arsllel cylThder re

asumed to he oscllatng vertically.

The coordinate

system is taken as shown in Fig-are-1,

The :rallel cylinders and

(j

are simulteneously beav-ng- with the amplitude Z,. Ohservia

_{the body © with the}

mirror effect, the wave transitte _{to the iositive y-direction}

consists cf the directly scattered wave and the reflected ;-Iave

rl 'Jhen , passes by the body the rélative motion between the body and the wave _{:.'ields the secondary}

wave s :, and The relative motion is assumed as;

Z2 ò[

dl'rl y=C

ìdl

'y=O]

'

where be expressed as =exp(-K) with diaft T.

Ir t o same way, the waves _{and are generated by}

the relative motions Z=í1

. suming the weve

Ç

yO

is replaced by the travelling wave at infinite distance, the amnLithde can be exrese.ed by _{=7Z with o'ovicus nctation}

an

of , and the phase ¿ by the method. as shown in ref( 1

Accordingly the final resultant

travelling wave is;

( 2

e,,=nE, -t-(n--)KQ

Using this ecjuation the

damping of the parallel cylinders N is calculated by a stripwìse integration;

c3

(3)

where Q should be smaller compared with the length L and the wave length corresponding to w

On the other hand if, without considering the interference between

the

reflected waves and the body, a

simple superposition

of the two waves is apilied, tie amplitude of the resultant

ravelling wave is 2Z,cos(KQ/2), and the damping in this case

becomes to;

o

2 -2

N =4cos ()

_f

A dx

w' L

Defining from equation ( 2 );

'wa/2COS

ation ( 3 ) becomes; 2 2

'=cos2()

f 7'

dx C ' w1 L resulting to; N _,2 2 A _dx/f dx c L L

7)

The factor f. indicates the hydrodynarnic interference which shows a speciality of the catamaran.

On the

contrary

2co5(/2j can 'se called the geometrical interference.

Figure-2 shows the calcalated damping

coefficient of a catamaran model witn a theoretical form using equations C )

and ( 3 ). The included value f is shown in Figure-v.

(6)

-.

RiLTIûN BETT.

ìV-XOITI

FORCE kND DAPITG

A relation 'OCt\veCfl -the heave-exciting force by t'ne wave on a restrained hull and the damping is described by

Llaskind-Newman as;

za

where Ç=wave amplitude

F =wave force amolitude za

i=wsve direction

In case of the catamaran, as the effect of the float distance upon the dampin ha been expressed by a factor

as is shown by equation (

7 ), the ec on the exciting force

can also be represented by a factor f, as;

N(w, Q)N(w)f.:( )

ca' ttF

Q)

where N(w) and P(w,;) contain the geometrical _interference

term.

According to equation ( 2

N' (w,Q)_

_{'w,;t, Q)2d}

C ca

and toether with equation ( _{9 ),}

{(

2

1_{- I} is obtained, resulting to a possibility to calculate the factor

T' from

f1.

An assumption is applied to c8lculate the wave exciting

force that a two-dimensional re1aton oy _{ewman (ref 2);}

za K 'a 12

is applicable to -the strip of' _{a hull in oblique waves. Since} the wave force with the amplitude given by equation _{( 17 )} is;

F =F cos(wt-Kxcos-rE,)

( 1

z za

with the origin at midship and the x-axis longitudinally alcn

(a)

io

}

the sb o, ikir the geometrical interference 2cos(Kin/2)

ir,to nccourit and assuniin a longitudinally symmetrical bull,

the total force amplitude becomes;

,1( $v

-=

cos( 'jf cos(K:co)dx

( 1.1

L T

rom euationo ( 9 ) and. ( 11 ) the F1 including hydrodynamic

interference can be calculated.

iore over, comparing the value calculated by ejuation 10 ) with that of ejuation ( 3 ), the reliability cf the

u:ìttiors anì the calculation is checked. Some examples of the calculation of the wave force on the above-decri'ced model are shown in the upper halves in Fijares-4 to 6. The

lower parts in the Figures show approximately calculated heav-ing motion applyheav-ing the a'cove-deccribed force. Eere, the added mass io assumed to have the twice value of the unit float.

A (!'T rT1? r r e'n mi;'

t'(TT

rn

%.1 ..

It is natural that if the exciting force is zero the motion is zero too. The fact seea;3 to be applicable also

even, in case of zero-damping from the calculated. result. The moot interesting case is that when the zero-damoing frequency coincides with the natural heaving frequency.

otora and IToyama (ref ) have shoim a similar reiult on a

unique section form from a view point of zero-heaving force. In this case, however, it was impossible to let the zero-damping frecuency coincide with the natural heaving frequency. Iad.

enly it was shown that a high peak of response appeared when the damping came close to zero in the vicinity of the natural freiueney. The conclusion seems to be apriicable to the catamaran from Figures-1 to 6.

Ir, case of the catmaran, it is cuite easy to let the

natural frecuency coincide

with the zero-damping freency.

With this iaode1, the

calculated.

natural frequency

7.63

sec1,

corresponding to the zero-damping distance /L-0.265 . In the

transverse waves, for instance, this catamaran has no damping and does not receive heäving force principally. The calculated

result in thi.c ease does riot show a:v iigh peal: as Figure-E.

th case of 'aneverce waves

(9Oo)

the zsc--e:citinj force frcjuency does not neccssarii coincide wIth the ro-da: ping frejuoncy. This results fro: the difference of the eocetrica1 interference term, ie in the force the

term is counted three-dimensionally an in the dani1ng

two-dimcnsicnalij. By tho sarne reason, the Iaskind-ewman relation is not fulfilled in a freuer.cy range including zero-damping frctenc,', notwithstandin fairly good agreement in other

frejwncy ranges.

In Figure-7 an exa.;iirle of the test result on another

catamaran model is shown. The ±'reuency is converted to that cf 2 rn-length model. Though the model was not restrained to .ny direction of motion, the results of the heaving moticn

seems to represent a simi'ar cnaracterstics to the calculation.

. CONCLLT3IoNC

The most severe problem for the catamaran is the relation between the natural frejuency and tite float distance. It

looks better to avoid the distance with high peak as shown in

A 4 ' a-4-

i

gr--1-

o e _

The similar treatment will be applied to the pitching motion, though a test result (ref ) has shown less evidence

than the heaving motion. In case of the rolling thé wave length of interest become half of that of the heaving, and there happens not so severe problem ±n the sense considered

with tI-ic heaving motion as far as the float distance is restrict-ed as generai in

Q/L<O.5

6. CONCLUDING RiiiJKS

since the relation between the float distance and the natural period is understood to be proper to the ship herself,

the acove-mentionea conclucon could be applea also to the

catamaran with advance speed.

The calculations were carried out on a digital co.tioutor

.,

J. ₎ . Oawa "The Driftin& Force and Moment on a ship in

Oì1cjue _{euiar :Iaves" I..P. Vol.14, No.149, 1967}

2 ) J. Ç. Newman "The ¿xcitin- Forces on Fixed Bodies in

taveo" J.J.R. o.3, 19b2

( . ) . totora ano. T, oyarì "On Wave-xcitaton Free 3hip

Forms" J.L.IT.A.J. Vol.117, 1965

4 ) .. Oawa "Some Model Tests on the Seaworthiness o± _a

Part-2 (ranslation)

T1i2 RLTIOT iJDL?D i"iA _{ND DAMPING}

C' .. CATA'11RAN

A. Ogawa

In case of forced heaving motion, a relation between

added mass and damping is hom by Kotik as ìramers-Kronig

relation (rs );

p'(w)-p'(oo)410p(x)_cdx2 (

o

x-w

The parameters representing the added mass and the damping

'(w) and p(u) can be replaced by in and N in the equation

m z z

of motion resulting to;

n QQ

-g,w)zri,w)-xnoo)ç ._(x)

- I.

/ f / - ir

X -W

Allowing the same relation to the

_catamaran;

_) 00

g

(w)=m'(w)-m'(oo)= N'(x)

C C Ti C -

L-O x -a1

Since the m(w) and _{(w) are known, if the practicability}

of the equation ( 2 ) is cnrtified, xn(w) can 'se calculated

from

(w) using equation (

_{3 ).}

iu-i interference factor of the added mass will be expres-sed by;

n'(u'\i) rn'"L- (w) in (w)

m z

analogously to N, and z

The practical calculation is carried out as f showing;

p

2-1

C (w)= =1

O :_wL

(1g ::(;)

_:

:::(x)

uj±& N (x)

;

,a-+-J

L L L L W

)ItUJ

O

x -w

x -w

w-S

w-g N

(x)

__

N (x)

j

Ur

:

(x)

.

Ix+J

dx

L L L LW -rW O X -W W+o

(x)

-dx

¿W L-Ui

Thefourth integral i

rewritten

by

replacing

x-w=

as;

I \ f _-.\

-) W+& X) S

dx= d3

L

.tere, if an approximation

is applied;

C4

The

former three integrals can

'be

integrated directly

numerical-If another approximation for ecjuation

( 5

);

(w)=ap4 ±b3+c2+d

e

is applied;

and;

The coefficients in ecjuat±on

( 6 )

are expressed by;

/4 7

,(w-

¿)=

a

-

.ó+

cc5

- d&±e

N (w)

= e z

N(w+

&)=

b8--

d6-re

N(ui+28)=lGa4+64c2+2d+s

i

2-2

f ,-ç C-1 ₍

r

, r

'

-¼.4

kj.1

W+8)- (w-s)

În execution of the calculation showed

that;

(u,)=m. (w)-m () > -O

z

> -O

and that a value;

du.i)=m0(w)-c(w)

was almost constant in the whole frcquency rance referrinc to

m0(w) which was directly calculated for the _{unit float, chowin}

an accuracy of the calculation of

Accordnly, an assunpticn as;

will be permitted.

In case of catamaran, as the effect of _{/L is nccleced}

at w m'oo)=2m(oo) can be accetea. _{Conciderznc t.us}

' and apl:lyinc oc1uation ( II

) the added mass is calculated by;

m(w)=

(u;)+m'(o) ¼, C C

=(w)+2m (oo

sO -c ±ir (w-28)--N .7

Aroper value of w1 can be taken in _{a larccr frequency ran}

The attached ficures chow the calculated

_5amin

coofficiont, added mass and c and curves, and f factors

REF ILNCE

)

J

T'j

and V. Man1is "On the

_{a:iers-Kroni:'- Relations}

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