A General Form of the Interface Boundary Condition of Fluid-Structure Interaction and its Applications

CHINA SH!P SCIENTIFIC RESEARCH CENTER

A General Form of The Interface Boundary

Condition of Fluid-Structure Interaction and Its Applications

Wu Yousheng W.G. Price*

December 1986 CSSRC Report English version-86010

( Presented at the Shipbuilding of China, Selected

papers of the Chinese Society oí Naval Architecture and Marine Engineering No. 1, 1985)

* Departement of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, U.K.

P. 0 . BOX 116, WUXI, JIANGSU

A GENERAL FORM OF TH INTERFACE BOUNDARY CO1DITION OF FLUID-STRUCTURE INTERACTION

AND ITS APPLICATIONS

Wu,Yousheng CHINA SHIP SCIENTIFIC RESEARCH CENTRE, CHINA. Price, W.G. BRNEL UNIVERSITY, UXBRIDGE, MIDDLESEX, U.K.

Summary

A general mathematical expression of the interface boundary condition describing fluid-structure interaction

in three dirensional hydroelastic analysis is presented [1,

2,3]. This interface condition is applicable to an

arbi-trary shaped floating flexible structure which nay be tra-velling or stationary in a seaway.

then

the motion of the structure is rertricted to allow only the degrees of free-dom of rigid body modes thi3 general interface condition reduces to the well known Ttznman-Newman relationship

develo-ped in sea}cEeping theory [4). If the forward speed of the

travelling tructure or the speed of the steady flow tends to zero, this general interLace condition simplifies to the wetted surface boundary conlition

imposed on a

fixed

flexi-ble structure ccrnnionly adopted in the literaturesL5-91.

Under the assumption of linearisation the proposed interface condition may be used to solve the potential flow problem

surrounding a flexible moving body, and hence ta formulate a corresponding boundary integral technique. Combined with

the modal analysis of the ìry structure, i.e. the principle of the modas superposition, the hydroelastic equation of motion of a three dimensional floating structure may oe

established and the rigorcu3 expressions for the hydrodynamic coefficients and wave exciting forces consistant to the

finite elerririt approach for the dry structure analysis be

obtained [1,2,3.. A selected set of examples of the nume-rical results are shown tc illustrate the applications of the present theory.

-1-Introduction

iydroe1asticity is tnat

branch of science which is concerned

with the phenomena

involving mutual interactions

among inertial,

hydrodynamic and elastic

forces.

The term 'hydroe1asticity"

first appeared in the technical literature in

the Spring of 1958

c1oJ

Mutual interaction between different types of forces

arising from the motions

of the liquid and a

deformable body is

the necessary condition

for classifying a

problem as one of

hydroelasticity.

This kind of interaction exists in ships and

offshore structures, as

well as extensive types of other

engine-ering structures.

o matter what

hydroelastic problem is, it

undoubtedly embodies the

full complexities of

the dynamics of

fluid and structure

considered individually, and in addition

special difficulties

associated with their

linking through the

interaction forces.

At the early stage

of the development of

hydroelasticity most of the

work was concentrated on

analytical

approaches and was concerned with the

acoustic radiation and

scattering from a elastic

structure with simple shape

immersed

in an infinite homogenous

medium (for example,

Eli] ).

Duririr

the last two decades

several numerical techniques, the boundary

integral method, finite

element method and the

mixed approaches

were

developed[,6,9,12-14J.

For a flexible marine structure

travelling in waves the

effect of free surface and

the forward

speed of the body must

be accounted for.

During the late

seventies a hydroelasticity theory tackling this

dynamic problem

for slander ships was

developed using the combined,

strip theory

and Timoshenko beam theory [1 5].

Recently the same idea implied

in this two dimensional theory has been extOEc.

a more

generai three dimensional

theory for a fiexiole

body with arbirary

form which may be

floating and moving or

stationary, or it may

be fixed to s.eabed.[2,3].

In this theory the

fluid motion around

the body is analyzed by a potential theory, and tne responses

of

the bodily motion and

distortions of the structire are

represented

in terms of the

principal coordinates associated

with the

principal modes o.f the dry

structure.

If the historical

back-ground of the two parts

of the problem is examined seperately,

it can be seen that the

three dimensional potential

theory

of hydrodynamics has been

well developed for the

fluid motion

around

a rigid body in

the seakeeping theory [16-18], and the

modal analysis of the dry structure has had no difficulties by means

of a suitable finite element

approach c19,2O. Therefore one of

the major tasks of the

three dimensional

hydroelasticity theory

is then to establish the more

general form of the interface

boundary condition on the

wetted surface of the

flexible structure

and to extend the

existing potential theory to the one for the

fluid motion around

a travelling flexible structure.

This

is described in this paper

togather with two examples

demonstra-ting the applications

of the boundary condition

and the relative

three dimensional

hydroelasticity theory.

-2-Structural Dynamics

Suppose that the dry continuous marine structure is desc-retized as a system with m degrees of freedom in a finite element approach, arid the structure i assumed to be linear, the generalised linear equation of motion of the structure may be

a (t) + b (t) + c p(t) = z(t) (1)

In this equation the principal coordinates associated with the m principal modes of the dry undaznped structure are.

rep-resented by the mxl column matrix p(t). a, and e are the

principal mass matrix, damping rnatix and stiffness matrix of the dry structure respectively. They may be expressed as

a = DTMD b DTCD, c = DTKD

where the mxzn square matrices M, Ç and K describe the system mass, damping and stiffness properties respectively, and the mxm square matrix =

1'2'

is the principal mode

matrix of the dry structìre. Ailmthese matrices are associ-ated with the relevant quantities defined at all the nodes.

The generalised external force acting on the structure arising from the waves, the

hydroiynarnic

actions and the mechanical excitations etc. is represented Dy the mxl column matrix

z(t )=jz1 (t),z2 (t),... zm(t)

Once the principal coordinates are solved from eq. (i), the components of the deflection at any arbitrary point defined in

the equilibrium coordinate axis system Oxyz may be expressed as an aggregate of distortions in its principal modes

u

(u,

V, w) = _r

_Pr(t)

(u,

Vrs w

p (t)

₍₂₎

r1

r=1

r

r

Here Ur denotes the r-th principal mode shape vector associated with the dry structure, which may be expressed by a suitable linear transformation of d , the r-th mode shape of all the nodes within one particular finite element, namely

T

u

i

ILd

(3)

where N is the appropriate shape function adopted in the finite element representation, L and i are the trasformation matrix and the submatrix relating the local coordinate system defined in the finite element to the global axes system,

respectively.

-The rotation vector at any point is

O(x',y',z',t) =

(4) where e

X',',Z')

= (O ,c ,R ) = X y z

-r

r

r

r

w r

r

r

r

-1---

-)i +

(-s2yI zI

-

zt

For a marine structure acted by the fluid pressure p only, the r-th component of the generalised external force is given

by [2,3)

Zr(t)

₌ T

s

where S is the wetted surface of the structure.

r

r

+

The Potential Functions And The oundrry Conditions

When the structure is placed into the water the fluid motions are interacted by the moving structure. Assuming

the fluid to be incompressible, inviscid and irrotational, the fluid motions around a flexible bod;ï may be specified by means of the linear velocity potential

z) +

'I' (x,y,z,t) i

(x,y,z)

+

(x,y,

D

(x,y,z)

+ E

r

(x,y,z)

J e

e

r=1

r

m

iwt

(5)

The linear potentials , 4 and _D describe respectively the

steady motion of structure° in calm water, the incident wave and the diffraction wave. The radiation wave potentials 4r

(r=1,2,...,m) correspond to each of the principal mode of trie dry structure.

These potentials satisfy the Laplace equation, the linea-rised free surface boundary condition, bottom condition and suitable radiation condition at infinite distance from the oscillating, translating structure 3,16J,.

On the mean wetted surface area of the structure the

potentials and

D satisfy the condition

=

- -7;- (6)

where n denotes the outward drawn unit normal vector of the wetted surface of the structure.

Over the instantaneous wetted surface area S of the struc-ture the potential satisfies the kinematic condition

=V .n

(7)

-s-where

U!

-denotes the local velocity on the wetted surface S.

When only the steady motion is considered the velocity of the steady flow relative to the moving equilibrium frame of axes is

= UV(4-x)

Substituting these equations into eq.(7) the boundary condition on the instantaneous wetted surface is

-5-=

-

1.n

)ri

-Due to the motion and distortion of the body the position of the wetted surface of the body is continuously moving.

The

flow velocity associated with the instantaneous wetted surface may be expressed in terms of the steady flow w, and a

vari-ation due to the displacement u to a first appoximation as follows

= (1 +

u.V)WL

_s

-

_s

In a similar way one has

= (

+ uV)V_

To a first approximation the variation cf the unit normal vector of the body surface in its unsteay state position S about that in its steady state position S due to a parasitic rotation may be written as

= ₊

s s.

Using these relationships and neglecting the second-order

terms in,

u and the boundary condition (B) may be of the

form

= + exw - (u.V)W].n

an-

- -

-

-

-This equation is valid on either S or S, the difference being of the second order.

Assuming that trie temporal variation of the untesdy mot.oris is sinusoidal, the principal coordinates have the form

iuJ t

= pre

e

(io)

with amplitude p (r=1,2,...,zn). Substituting eqs.(2)-(6 and

eq.(iO) into eq.9) and considering the fact that this boundary condition must be satisfied for any arbitrary combinations of

p , the wetted surface boundary condition

therefore has the e8uivalent form for each

_r

= [iw u

+

- (u .V)W),n

on s

for r=1,2,...,m (li)

an

e-r

-r -

-r

-

-Eq.(9) or (li) is a gener9l form of the interface corìitior. of the fluid-structure interaction.

To adopt this boundary condition the steady motion ir1 calm water must be initially known. This significantly increases the complexity of the solution of the radiation potentials. The simplification conmonly accepted in hydrodynamic inaiysis

is to assume a special case where the perturbations of the

-6-ons

(8)

- 7-..

stesy flow due to the forward movement with constant velo6ity U of the body are small and negligible, namely

= -U

In this situation the body surface boundary

condition

reduces to

-

iw (un +vn

+wn)

3n e

rl

r2

r.3

a aw av au +

-[n3--4-

i-4)

n2(4-

--))

r = 1,2...rit (12)

fl-xen e speed U of thesteady forward motion

or

tne structure or the steady flow is negligible eq.(11) becomes

a

= it

u.n

e-r-This is the formulation widely adopted in the works of fluid-structure interaction by other authors L5-9).

If only the six

bodily motions of a floating rigid body are investigated, namely surge,r=1; sway, r=2; heave,

r=3;

roll,r=4; pitch, r=5 and yaw, r=6, eq.(9) simlifies to

= + ÇxW - (c.V)W1.n

on S, S

n

-

--where

a=X-X'=r+xX'

is the local displacement.

= (x,y,z)

and x= (x',y',z)

are the coordinates of the point under observation in the equilibrium reference frame Oxyz and the body fixed refereflce frame O1x7yzt respectively.

iw t

_e

= (p1,p21p3)e

iw t

= (p4,p51p6)e

e

are the unstesdy translation and rotation of the body.

The relationship

(13) was derived by Tirr.xiian and

Ìewman [4J in the development of the mathematical model to describe th rigid body motions of a ship moving in waves. With the restriction that the unsteady motions are sinusoidal in time with the frequency of encounter w , the boundary condition

(ii) governing the

radiation potentials for

a rigid

floating

body becomes the well known form [i6J

and

G±(r,r)

=

G(r,x1,y1,z1)

G(r,x1,-y4,z1)

Accordin

to this definition,

(represents either

or

Ç ).

ç, and « are

the composite potentials, source

strength and

dipole strength respectively.

Three (not all) of the

composite

singularity

methods may therefore be

expressed as follows:

-

s-r

_{1w n + Um P}

_{(r=1,21...,G)}

et

r

r

(14)

for r = 1,2,....,6 and

(n1,n2,fl3)

(n4,n5,n6) - Xxn

(rn1,m2,t3) -(n.V)W/U (m41m5,m6) -(n.V) (XxW)/U

Potential Flow around A Floin

Flexible 3ody

The main

differences between

th

potential flow around a

flexible body and the potential

flow around a rigid bod:r is

that the former

has more components. Each

comoonerit depends

on the

correspor4dng flexible body

mode u

( r7 )

and governed

by

the

correspondino interface boundary

cBndition (li)

or (12).

Once

the

r-th mode 3hape of the flexible

body is knon the r-th

component of the pc;entia

my co

aetermined by employing

eq.(11)

Rnd a

suitable

boundary integrai technique

(for example,

one of the singularity distribution

methods) provided the steady

flow velocity w has already

been solved.

The detail nu,ierieal

techniques adopted may be

found in the cited references

by the

end

of this paper.

For a structure with port and

starboard

syrnrnetry, as it

always is in the conventional

ships and offshore structures,

an

efficient composite singularity

distribution method may

be used

to evaluate

the unknown potentials

associated with the

rigid and the flexible body

motions [2

Asswing r = (x,y,z) and

= (x

,y1,z1) are an observation

point4 and a singularity point

respectively defined in the equi

librium

coordinates system, f(r) is an

arbitrary function and

G(r,r) is the Green's function, the

corresponding composite

functions are denoted

by

(3)

Composite nixed method

4

11It

_ìn(r)

PGdS + --2Uiw4

-

U2__]GL

,2L G1

dc

for rCVUS

e cx1 i

-+ p- on S

_p.

+

2 ±3

+ U

--JnuC

-9-4 = ;j:-

Jj(r)

+ ;j--

-tor rCVUS.

(15b)

+

In these expressions P

are the composite bo.mdai'y

coriditior.s:

pi

_{(x,y,z) ±}

When = ,

it gives the coraposite form

of the interface

condition of the

radiation potentials, namely

+

P _{= 2L} U + O xW -

(u V)W}.n,

e-r

-r

-r

P

=0

for syTrrnetric

odes, and

o = +

-I

j

}

(15c)

(16)

(17)

(18b)

(1)

(2)

Composite

+ +

e-

(r) =

_4u

Compo3ite

sorce distribution

dS

nethod

4

QGfldC

cV or S

p

metrod

(15a)

Q (

+- J

C p

dio1e distributicn

for antisymmetric

modes.

The Green's

function G(r,r. ; pre-satisfies the free surface

boundary condition, the botton condition

and the radiation

condition at infinite distance

from the body.

The composite

source or

dipole

strengths for the diffraction or the radiation potentials can be solved from eq.(16), which is then a boundary integral equation defined over the

port

part of the body with the volume VP and wetted surface Sp. The advantage of this method is that it requires half size of tne descretized matrix equation and much shorter O? ti:.e

in comparison witn

the

ori-ginal singularity method where no port-starboard symmetry is introduced.

Generaìie

Equation Of <otion 0f The Hydroelastic System

After the er;pioyment of Lernoulli's equation and the

neces

sary manipulations,

it

may be shown that eq.(1) may take the

lwt

(a 4 A) j(L)

+

(b ± E)

(t)

4

(C + C) p(t)

E e

e

= E(t)

where A, B and C are the hydrodynanic inertial matrix, damping matrix and restoring or stiffness matrix respectively with

their components being

A=

_rk

neJfn.0 [iw

+ w.v]us

j

r

e e

= -

£_ 1ml n.0 [ice

+ dS

rk

w

j--r

e e s and

Ck

_2JfflrtWk

+ k 21d5

(22)

The second tern in the inte;-rl of eq.(22) arises from tne restoring

effect

due to unit

unsteady motions

of the structure within the tedy pressure

field.

For

general three dimen-sional arbitrary shaped ooJy this term is cf

the same

order a the first term. E is the column matrix of eneraljsed wave

(20)

for nrtisyetric

flodes.

The corosite source or dipole strengths for the d±ffraction or the radiation potentials csn be oivea from eq.( ), which is

then ntegraJ. eqution defined over the port part of the body

with the vo1ure V and the wetted surface S The advantage of this method isthat it requires half siz of the descretized matrix equation .'iuch shorter C?TJ time than the originnal

in'u1arity method without introducing the port-starboard

excitir» forc' whc3e r-th cor.onent is

=

PÍJt!.ur

To a first ar,roirnqtion these e,ressior.s are rtiona1 and rigorous. They are also consistant with any kind of finite e1e-ment aproechs employed in the dry structure an1ysis rovided

eq.(3) is zubstituded for

u.

Especially when r.ae1y

the structure is slender, thin or flat, or the forward speed of the body is small enough, these hydroynnmtc coefficients and the wave exciting force become

rk

4

Re(T)

w e R

rk

fi)

rk

, e

Ck

_2PgJJr.wkcìS

T

= -2ç

_J(tl

&!!:'- (&3dk)S

ecS -e and

= P

JICT

UJ)+

(ii-U-)JdS

ecS

-e

fer syrvnetrtc modeg in rr1,2,...1m

rf

(TTNL

X - ta)

e x k

ecS

c foi

ntiynmetric niodes tri r=,2,...,ri,

and , and are the composite potentials of incident,

diffraction and radiation waves respectively. is the wetted surface of an element. The sumation is taken ov&r all the

elements on the mean wetted surface of the port hull

-.

12

-(23)

(fT'r

ti + E

(n .ÑLd ) (i

- U')ç dS

-n

ec 5 -

- --y

e ix k Where T

rk

E.(19) may also be used to predict the forced vibration of the structure travelling in cairn water due to mechriical excitatiOfl3. In this case only a slight modification should

be made to the generalised force F in eq.(19). Asstmting that

a sot of concentrated exciting forces

tjet(i=1,2,...)

act on the body at the positions

(x,y,z)(i1,2,...) respectively,

the gAneralised force, replaing that of eq.(23), can be

ritton in the vector form as

- - ii,t =

_4'

_{'- jr-'1'J.''].}

(.,.

4 - 13 (24)

E.(19) describes the motions of a non-conservative system with The matrices A and being neither syrt.metric nor positive defir.te but frequency dependent. Although this equation is given for siruasoidal excitations it mey be easily rnodfied and extended to include other linear or non-linear external loadirgs, for example, the influence of a mooring system, or

the nn-linear fluid actions. The principal coordinates

p(t) may thus be deternined from this equation, and the steady sinuasoidal resronses of the displacements, bending moments, shearing forces and stresses t any point in trie structure be predicted from the relevant modal superpositions.

umer±cal Aoplicatons

To il1ustrte the aoplications of the present theory a typical set of calculated results are presented for subrner ged spheres and a SWATH ship travelling in head seas.

The fThw due to two sheres moving in unbounded fluid

or crie sphere movir near a Wr1I is one of the classical rroblems of fluid-solid interaction. It analytic1 solu-tions can be obtained, for example, from [21] and 22]

In order to validate the nresent theory and its numerical techniaue the hvdrocvnamic coefficients of two spheres, of

radius R im, oscillating with frequency ù in parallel or oeposite directios, as well ac one schere moving towards a rind wall were caiculoted. Fig. shows a mesh of 224 panel elements used for describing the wetted surface of the

spheres. The numerical results for the deeply submerged

spheres ( in this care z =50R ) ow that the values o1 the added mass are freouency°independent, and agree with the

analytical results for spheres moving in unbounded water or by an infinite rigid wall very well, as shown in Figs. 2-4, where the variation of the calculated results with the

dis-tance between the two sDhere centres are demonstrated and compared with the curves given by the analytical expressions. The maximum interaction distance between two moving objects in this case is about 5 times of the charateristic size of

the body.

When a monohull flexible structure with typical two dimensional profile ( for exrml?, slender uniform ship

ws analyzed by the oesent

theory,

the

results were quite close to thore given by the two dimensional hydroelasticity

theory (the strip-beam theory) [2,3,15] , as might have been

expeced.

For a structure whose form is far from beam-like

the present theory shows great feasibility, as discussed elsewhere [23,24]

Fig.5 illurtrtes a hypothetical model of a SWATH ship, te distribution of its mass and stiffness, and the mesh of its men wetted surface. The principal modes of the dry structure is analyzed using a finite element presentation which consists of 32 auadriltral facet shell elements,

10 eight noded facet shell elements, 12 eight noded thick facet shell elements end 32 beam elements with offset.

A few selected numerical results of the added mass end clamping coefficients for the S4ATi travelling with Proude number = 0.223 are shown in Fig.6.

When the SWATH ship travellinp, at Fn=0.223 in calm water is excited by an exciter situated at the stern of the port hull, the principl coordinates calculated from ens.(l9) and

(24) show typical feature of reonncs, as was expected.

-Narrely, esch principal coondinate has one doriiinant peak at a so clled resonance frequency. The resonnnce frequencies

nd the natural frequencies of the dry structure are given ir. Table 1.

Theee resor.ences are excited once again by the wve actions when the S'.AT ship is rvelling at 12 knots (?n= O.22) ir. sinusoidal

regulr head wves.

1n this cs

the

enerq1iced wav exciting forces for yrtetrie ìodes r= 3, nd 9 ere illustratert in Fig.7. h'

comarqtive

rnitude of the princioi1 coordin9tes ocitod with the svLr'etric distortions re given in Fig.S.

The orinciol coordintes

p disolav peal correspondin to the r-th resonance

Cre-auncv and other rescnance re'lecting, the coupiir between the different distortion

ode.

addition they display peaks at ui/L/ = 4.7, 2.2 and 2.T' hich are coincident with

the peaks nnebrin in he wave e :tinr forces of Fïg.7, and are believed to reflect the occurnce of the locslisd

stan-d4r'- ve

heroc' betw"er

he ruts of the gWATJ-! structure.

'i.9 il1ustrate

the a:u'e ':' the bendino roent

alor the ler.th of the pert he variations of the

ajun strese' c

:he outer sur:'T e of the rort strut alonR

the joint with h':I..

t is evident h:

t'

na

o'

is efficient to

analyze the dynic stren:-n

md rtdict the positions where

a possible fatic'ue crck I kel to occur'e for c

rnrine structure resDondin

7".

in ddition the

effect of the fluid inter - :'r examnie, the localised

standing waves) ir. erhanc.m ::cturql loadings may be examined in detail.

Corr lus ior.

A 7eneral fori of the interfecr boundary condition for 1inkirr a three diesione!l flexible body snd the surrounding

fluid has been prsente.

ased cr the techniques of struc-tursi dnarnics and the rotential flc analysis, and the use of the rresented intcrface conditir a general unified theory of three dimensionnl hydroelsticity has been outlined for the evaluation of the responses of a mrine structure which may be either sttonay o' moving with constant forward speed

in regular waves. This theory is consistent in both struc-tura! and hydrodynamic aspects, seekeeping and distortion analysis and is sble o account for the forwRrd speed or the steady flow effect.

The examples of

the

rsu1ts show that bending moments and tresses (as well na disolacerients, shearing forces etc.

at ny positions over large rarine structure subjected to waves may he predicted and the justification of the struc-tural safety be made without the reed of making semi-errrnirical approximations.

Although the nresentation and

he exnies given in this

paper are based on linear theory and regular waves, with slight modificqtiona sorne non-linear fluid actions may be included and the epproacb be extnded to r irregular random seaway, hence

rnor rt±c redictions of long term structural

behaviour be pvided.

-Nomer lature

Coorcinate systems

Oxyz Trqlatinr Cartesian equilibriurn system with O on

the water surface

on the same vertical with the centre of gravity of the body, Oz pointing

ver-tclly pwards and Ox

pointing from

stern to

bow; i,

and k are the unit vectors

of the

three axis respecti'e1y.

O'xTv'z' ody fixed

Cartesian

coordinate system, which coin-cides with Oxyz when

the body is in

its riean

pos ìtion.

Generel convention

and

-

In'icate a vector and a natrix

respectively.

+ or -

(used

sup;rscrict) refers

to a composite

urct ion.

Symbols

'43

e Frequency of wave enc':nter.

Reference s

t. Bishop, R.E.D., Price, W.G. and Wu, Yousheng.

'A general linear hydroelRsticity theory of floating structures moving in a seaway'. (to be published,

1985)

Wu, Yousheng. 'Hydroelasticity of floating bodies'.

Ph.D Thesis, Brunel University, U.K., 1984.

Price, W.G. and Wu, Yousheng. 'Hydroelastïcity of marine structures'. Invited sessior'

lecture IUTAN,

Lyngby, Denmark, 1984.

¿t. Tirnrna,n, R. and Newman, J.N. 'The coupled damping coeffi-cients of asymmetric ship'. J. Ship Research45, 1962,

1-7.

Zienkiewicz,O.C. and Newton, RE. 'Coupled vibrations of a structure submerged in a compressible fluid'. Proc. mt. Syrnp. Finite Element

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Assoc.

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ulti-huli vessel

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March, 1935, paper 15.

Table I 'atural frequercies of the dry u1l arid resonance

frequencies of the hull trave.ing at

Fr=O.223 n aer for the SWATfi model in rad/s.

Index o dorinant dry mode 3 A s 7 9 10 12 13 14 15 16

Ca water rocra::ce Natural frecuency of

freuency the dry hull

0.65 0.517 0.237 7.317 5.057 -). i 10.757 11.447 14.727 1 6.467 :0.097 21.807 42.747 20 -9. 5233 9.6693 12.2737 16. 7 223 19.6402 19.7510 2..4474 35. 2605 36.0560 58.3713

5 O!1 y

rig,

_{Pan1 e1rnent description of two spheres}

and the axis system.

-5.0

2.0

1 2 4ry _3R3) 3 Jo -

₍₁₊

3 R3 -3 _16y3 o

(z/T5O).

3.0

o present results

2.5

7/

3 E3y 5 10 20 y0/R

50.

100

FIg. 2

Added mass of two spheres osci1iatng alQrlg

the line passing throuqh the centres (z0/R=5O).

6.0

o preer't results

-4.0

1 2 5 10 20 y0/R 50 100

FIg.

3

Added mass of two spheres oscillating parallelly

2 S .10 20 y0/R 50 100

Fig.

4

Added masq of one sphere oscillating by and

normal to an infinite ia1l (z0/R=50).

22

-22

4.0

o,

present results

4rr _3R3

5.0- \

A22 pR3

LCk Lumped au bulkheads

23

-7 3 . 2a t. C-o 52.12Erci -r L

::í

-

'I

r-

f N' I

r"

---r-r"ff---:

_.:Et;

4.541m 2.189m

5(c)

-J 4 .225m

r--

Bridging structure

-.---1___ _J -

_._._J__-_'1 One hull One strut r

Fig. 5(b)

o-s rc/t. t (Hu.i3j (4>2) 18. 326a 22.7 54m

'32

ne1 eiern'n-

EI(ridginq Structure) (*J)

ts.

L_J

EIringstruCture)i

EI (Strut> (t.OzsI -,__.4 EIStrut) ()*Lm)

T

H

_L

H

Li

1._i

rin. 5

A hyoothetic

ödCi of

ì

ship:

tuctur1 roei,

mass dist'ihution,

stiffness

ditri-bution,

(di the reh of the

wett6d surface

with a toa1 of

J\thed mass

nd

damping

coeffi-eient

for the

AT

model

aveI1ing at

'r= 0,223.

e

24

-tig. 6(a)

A77/(QV)

ta -i-e -

j &

w--

---Fig. 6(b)

w. sic Vg) tei t. I. e

Fir.

()

77/(pVg/)

Fir. 7:)

1F31

/(rr7)

3.! 3.1

t

- -s

Fi'. 7(.)-(c

The

eneriie

wave exciting

forr'p

för th

t

JATK mode?

trcve11ir'g nt

F= C.22.

o.,

25

-.. .

e -, u U

Fig. 7Cc)

FQ1

/(apgv)

Fig. 8(s)-(c)

The amplitudes of

p

for the SWATH

model

travelling

at F"n=O.223 in head seas: (a) Comparison of

(r=8,9,11 nnd

14 symmetrIe

modes)

,

the

structural dn ping and fluid viscous effeet are ignored.

Ch) p , the

strutural

dam.-ping and fluid

viscous effect

are included,

(c) i9 ,

the

structural

dam-ping and

fluid viscous

effect are

inc.-luded.

li e t, te . U e e e e e e e e e e e e e ta... StL.'9) 26 r 3 r t! I

rt4

r 8.

Fig. 8(a)

r.

r-q t;

L

W. i I I J r- t4

-Fi.. 8(c)

Fig. Ç(e-()

The responses

the SWA'P mcel

travellir. .t

Fn=.223 ir.. heed

se.qs:

() Vrirtion of

the arnr1itude of

the bendiri rnent

M

!og the

length of the port hufl at three frequencie5. .4 a .a si Lii, i L t.8 t.e,

(b) Vrition of

the rnaxiriu

5tre-ses with

t three ,osttiona

ori the outer sur- _:

face of the port strut alori the

join with the

in hull. Fig.

9(a)

.7

/

,1 e C ) C

---.C('.

Fir. 9(b) -

27

-A r Si1( L.'q i 26

"ic. '(c) Vrriztion

rnexiurn strese on the outer surface of the port strut nlonr

the joint with the main hull at

three frcueneies. - 5.e72 r.Øs

--

S72 04

28

-/

-Z '-.

x (mi y Fig. 9(d) VqriFìtion of the direction of I - I

on the outer sur- /

,

\

7

face of the port j ()

strut 1örg the J

3olnt with the .251

--mainhullat

three freuenciez. -041 - rQ5 L672