Basic relations of strip theory. Part I: Radiation problems

80)085

TECH!NISCHE HOGESCHOOL DELFT

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAAATKUNDE LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

Deift University of Technology

Ship Hydromechanics Laboratory Mekelweg 2 2628 CD DELFT TheNétherlands Phone015 -786882 Jacek S. Pawlowski Report nr. 544 March 1982

BASIC RELATIONS OF STRIP THEORY Part I

RADIATION PROBLEMS

Contents.

Introduction.

Kinematics of the Ship Motion.

Kinetics of the Fluid Flow.

page:

1

4

6

Normalization of the Boundary Value Problem.

Governing Parameters.. 10

Simplified Equations. 17

Memory Effect's 27

6.. Conclusion. 31

Appendix A. A Comment on the Criticism of the Su-érposition of Potential and Viscous Effects in a. Fluid Flow as presented

in _C6 3,3

Appendix B. The Derivation of the Rotation

Tensor R. 36

Appendix C. Normalized Simplified Equations. 39

Appendix D. Transformation of Kinematic Quantities

from the Inertial System of Reference. into the System Fixed with the Ship 42

-1-Introduction.

The present report has been prepared as a part of a project

concerned with the use of the rudder for stabilizing the roll motion of displacement type ships.

The investigation of the roll motion has quite long history. In t'he modern sense it was started by the classical works of

Froude and Krilov. A present review of the state of the art

can be found in. (i] , and the major recent developments in

roll prediction: are described in [2.] and [3]

According to the modern comprehension of the problem, in a

brief statement, the roll motion is understood to be a non-linear phenomenon coupled with at least two other modes of motion, i.e. sway and yaw, and strongly dependent upon vis-cous flow effects. Since the publication of t4] attention has also been paid to the stochastic character of the motion

in a real seaway. A relatively recent account ofthis is

pre-sented in _[5]

The involved character of the problem has been the cause of several disputes. The often controversial questions which seem to be of particular importance for further discussion are focused on the two following aspects:

the possibility of superposing potential and viscous flow effects,

the possibility of neglecting nonlinear phenomena..

Recently, the validity of superposing potential and viscous flow effects has been questioned in [6] , in connenction

with the model of lateral motions put forward in [2] .

How-ever, since within the present state of knowledge viscous phenomena encountered in the investigation of the motions can be treated effectively only by empirical and/or semi-empirical methods, the superposition technique is imposed by 'practical limitation rather than a scientific priniple. On

the other hand the reasoning presented in _[6] does not seem to disprove the technique from theoretical point of view, see Appendix A. Hence it appears that the question concerning the validity of the technique should profitably be resta-ted as one of proper taking into account interactions between potential and viscous phenomena.

) In this report nonlinearity means the presence of effects

which are nonlinear with respect to ship displacements as

2

The approach to the aspect b: mentioned above is of high practical importance and as indicated by experience from

other branches of applied mechanics, e.g. t8 chapter 10, it

should be based upon the purpose the prospective mathematical model is*to serve. As has been indicated in (8) an adequate

linearmodel may by principle be .extr:emèy usefu for predic-ting the ranges of ship operation parameters within which excessive motion can be expected. However such a model gives no reliable informatiOn about the excessive motion itself and hence it can not be applied to assessing the motion characteris.-tics which are of interest.for evaluating the impact of a sta bilizing device.

In connection with. the application of linear models for the investigation of roll, the importance of couplings .wih yaw and sway was stres.ed. in [8] . It should be pointed out

though,. that the existing lir'ear.models of these motions, such as e.g. [9 ,

io) or

ifj,

represent as a matter of

fact alebraic relations in the frequency domain,, see e.g.

f12] , and hence their use in the sense of time domain

dif-ferential equations should be considered with caution.

In order to round off this rudimentary discussion the impor-tance of ship steady velocity effects upon the phenomena under consideration must be underlined.. The development of strip theory, see e.g. [13], t14] and [9], shows the rele-vance of forward speed effects for the proper assessment

of potential flow phenomena., and some viscous phenomea appear to depend strongly upon forward speed too, C3) . Besides the

results presented in _f153 suggest that a signficant impact

can be exerted upon the roll motion characteritics by a

drift speed.

The above remarks can be sunanarized by putting forward for further consideration the following issues:

deiption of the primary potential and viscous flow

ef-fects and, if possible, of their interactions;.

inclusion of nonlinear phenomena, especially those in-volved in restoring and exciting forces., and damping and coupling effects;

3

4) establishing of a correspondence between time and fre-quency domain descriptions.

The mention of exciting forces at the point 2) above reflects

the basic dichotomy of ship motion fluid dynamics which results from recognizing environmental disturbances, such as sea waves and/or winds, as the primary signals in the system. In linear

models the generalized forces produced by the signals can be

clearly separated from those due to the motion of the ship. The problems arising in the estimation of these two kinds of

forces are different, s.ee e.g. [16) , which justifies their separate treatment.

In nonlinear models the flow phenomena produced by charirrg en-vironmental conditions interact with. the phenomena corresponding to the ship motion. However both kindsof phenomena preserve

their identity if the exciting flow phenomena are defined as

arising in the absence of ship motion and the others as being

Lproduced in the presence of no environmental disturbances.

Then,the interactive phenomena are clearly distinguished also

since they must dpend on both the environmental disturbances and ship motions.

in reference to the issues outlined above the present report covers a systematic derivation of boundary value problems cha-racterizing the potential flow phenomena due to the ship oscil-latory motion as a rigid body, or more speciafically, it is

concerned with the near fie1dboundary problem for forced

oscillations, in the terminology of slender body theory., e.g.

[16]

Although this kind of problem has been devoted quite a lot of

attention in the past, see e.g. _f173 , _{116], 'EiJ, [19L [201 [21]}

[221,

1237 , 1241 derivations were carried out for the linear

case

only*)

which allowed separate treatment of the lateral and longitudinal motions, and the influence of drift was

ne-glected. Besides at least two different forms of the free

sruface condition havebeen used by vaiiousr authors, e.g. compare [16] and [213

Hence in this report and attempt is made to present a

deri-vation which includes nonlinear phenomena and drift effects.

The derivation is carried out in the time domain and special attention is paid to the physical interpretation of the

employed formaiizm. This formalizrn is based on asca1ing pro-Nonlinear effects in the iane oscillations of cylinders have been investi ated separately by several authors., e.g. [29) , [30]

cedüre which leads to the normalized govering equations,

c25j [26], and the governing parameter appearing in them

are considered as the small paraniters. in the asymptotic

expansion technique. One of the important advantages of such an approach is that it clearly displays similarity features of the problem,

_{[253, t273}

. In particular, the requirement

of the sthallnes of ship displacements, which in the past has been a cause of some confusion r283, is introduced in a uniform fashiOn, on the hull surface..

Cartesian tensOrs' of the second rank are frequently employed in the text and the meanin of the corresponding no.tation

can be graspedfrom the material presented in the Appendix B.

1. Kinematics of the Ship Notion.

The oscillatory motion of a ship as a rigid body is considered. The motion is observed in an inertial frame. of reference

con-stituted by the Oartesian system of three mutually perpendicu-lar reference axes 0.:, x , y , z

0 0 0 0

In this system the reference configuration of the ship/s hull is moving with the 1-brizontal steady celocity V and is

deter-mined by the reference. hull surface S... . .

-The surface S remains fixed in another inertial frame of

re-fer.ence of the axes 0, x, y, z which are respectively parallel to the axes of the 0 , x , y , z system. The coordinates

0

0

0 0

z,z are measured vertically upwards from the undisturbed

free surface., the axis x is directed forwards along the sur-face S0 and the axis y points to. port.

The velocity of the point 0 with respect to the system

0 , x

, y , z is defined as:

0

0

0

0

V

LL.4+

V

(4.4)

where denote the versors along the x,y,z axis res

pectively.

The motion of the ship in the 0, x, y, z system of reference is characterized quite generally by the relations:

5

one being the inverse& of the other, in which r denotes the

position of a material point of the ship at the time t

whereas ' denotes the position of this same point in the

reference configuration. Both r and r' represent radius vec-tors in the O,x,y,z system.

For a rigid body motion of the ship, considered here,it

fôl-lows that:

where r denotes the vector of translation of the ship and

is the tensor of rotation )

The components of r and 1 are the same in both systems

of reference and will be denoted as follows:

and: see Appendix B.

1000 \

0 0

- J 1-O

0/

( 0

_LvL5

10

A 3

0

f0

-b _rLs

PZ6-LA

0

'15 0

-1 o(b73)

F

F(t)

(4.2,)

-

(t3)

with asignifring the unIt tensor, and. p

1vj,i1,

deno-ting the rotations, taken in the indicated order,about the axes fixed with the ship at the point P and mutually paral-lel to the x,y,z axes for =pi I7 =TO. The particular

presentation of the tensor will be explained later.

The radius vector in the system of reference

will be denoted by , and the components of the vectors.

and iil be expressed in the forms:

F0= (.x0101z0')

j"

corresponding to the versors e1, I = 1,2,3.

The above description of the ship motion implies that:

}

6

it foilowsthat the space differential operators Inthe

systems of reference 0 ,x0,y0,z0 and 0,x,y,z are

identi-cal and will be denoted byV:

a

'XG

_'Xe

II

on the other hand the dependence of the time differntial operators takes the form:

('\

....(L\

7L

oiI

1't)

b'

2. Kinetics of the Fluid, low.

It Is assumed possible to model the flow of the watr

surrounding the.ship by an irrotational flow of the ideal fluid of the specific density equal to the density of the

water under the influence of the acceleration of the

gravity force g. The flow is described by the velocity poten-tial in the 00,x01y0,z0 system of reference.

In terms of the potential the condition of flow continuity takes the form of Laplace equation:

(4.&

0

7

where D denotes the fluid domain.

The forces in the fluid are determined by Bernoulli equation:

p.=-

[('

(V+t1

with p denoting the relative pressure in the fluid.

The boundary D of the domain D is assumed to consist of

two impermeable surfaces: the free surface SF, specified by the equation*:

= 0

_onSF

whereL:ç (x,y,t) denotes the free surface elevation, and

the wetted, surface of the hullS, defined by the equation:

Otherwise D extends to points of arbitrarity large values of

rI

. The impermeability condition on 3D can then be written in the form:

+ v

_{.v] L}

_XoIJoi)

Ll

= 0

and,:

L

-v.v]

o

A straightforward calculation shows that (2.5.) can be

re-written as:

-vc.v

ov..S1,

Besides it can be shown on the basis of (1.2.) and (1.4.),

r32] , that the relation (2.6.) is equivalent to:

(74

-

n.j,

-v

.=- 0

Opt.

5

the equation (2..3.) implies that the vertical component of the normal at SF directed into the flui.d domain is

where:

By means of the expressions (2.10) (2.12), the

equations (2.7), (2.8) and (2.9) can be reiritten in the form

of the following conditions:

8

with the dots denoting the differentiation ith respect to

time. The free surface elevation is governed by the dy-namic condition on SF, which can be found from the Bernoulli equation (2.2.) by equating the relative pressure to zero

for z=:

£()

+(%)Z]

05F,

The equations (2.7), (2.8) and (2.9) present some difficulty

since the derivatives of the potential which they involve must be evaluated on the a priori unknown surfaces S ortSF. This can be circumvented by employing the following develop-ments in power series:

L +

£ (E +

o(t)

(vj+ t[(/]

+

o(2)

-I -1. _(2.14)

-v..v+ ç [(-v1

L(-3+

(z.'1

-v.v()tfl

orL. Z.O

I

+ vcp

+ _L

(,Yj

_v

_{+ (c',}

)] +

_{o (ç9,}

Obl. Z and:

[i+i''-V

+ + 0(1 2)].R.

t:O

,

Se

where S0 denotes the reference configuration of S. The com

plete specification of the boundary problen for

re-quires. the initial conditions for t = 0 ô.rradiation

condi-tions for

Ir

I-,o° to be stated,

[33]

, in addition to the

equations (2.1), (2.14) (2.16),these however will be dis-cussed i.nconnec.tiôn with the the simplifications of the problem which are to be presented below.

For the estimation of forces exerted by the fluid upon the ship's hull it is convenient to express the pressure on S

by the values of the derivatives of taken on

Taking into account the equations (2.2) and (2.12) it is found that,:

+

(c1)+9z*

+1jv)

9-) -

C .

¶ç7)

[(I

-

(v

21

- o(i &F I2)

-3. Normalization of the Boundary Value Problem. Governing Parameters.

in solving the boundary value problem presented above it is expedient to, take advantag.e of the almost generally appli-cable features of hull geometry as well as to involve some assumptions concerning the ship motion. Most systematically this can be carried out by normalizing, the governing equa-tions of the problem with respect to spacial and time scales chosen in connection with the assumed properties, [25], r261. It should be expected that in the vicinity of the ship the

proportions of the huil have profound influence on the flow kinematics, whereas at some distance from the ship their

significance becomes less important and the ship can be con-sidered as an elongated source of disturbance in a three-dimensional flow-. It follows that two different sets of spacial scales should be chosen for normalizing the boun-dary value problem in thettwo regions.

However in the present rep tt only the properties of the flow close to the hull surface will be examined and hence it

is sufficient to employ one setof scales.

In order to perceive the meaning of scaling it is convenient to assume that the problem under consideration corresponds to a nondimensional problem, the "spacial" coordiantes of

which are denoted by 2QI2Q and

respectively and that

the spacial coordinates can be clerived form thm Ly means of

the relations:

(x01

(L0110

_(3.4)

and

i,L1i31d

The equations (3.1.) and (3.2.) justifiy the name of spacial

scales for the real variables L,.B and d, upon which the condition:

L,S,d..>o,

(3.3)

I

see e.g. ,[16j

,t181

, [2:2]

*1II)It

_{is assumed that a part of the fre surface of water belongs}

is i'mpOethtThe relations (3.1.) and (3.2..) allow the s.pacial

relations ó the nondimensional problem to be scaled

(transformed) into the corresponding relations of the original problem or vice-versa.

it follows that the equation of the reference configuration

of the hull surface S.:

-transforms into the equation of a Surface in the non-dimensional space:

(3.5)

if it is assumed that:

Conseq.uentIyby choosing S0 in such a way that:

WLA.)(

yflX

,'re S.

and:

p1Y

-4

the scales L,.B,d are made to correspond respectively to the length, breadth and draught of the. hull surface in its reference configuration S0.

The importance of the scales results from the fact that in

normalized: spacial relations of the problem they form non-dimensional so called governing parameters,. r2'51 , which

determine similarity relations for problems, r2'51 r27)

derived by scaling from the same nondimensional problem'. This can be. illustrated by considering the vector on

in a normalized form. From the equations (3.6) and .(3.2) it is found that:

12

-It is assumed that the derivatives

Se

S _and

are bounded and continuous, on S , and that:

0

and/or

oo

On the basis of (3.9.) the normal vector nncanbe expressed

in the form:

tr

E2(9

)t

0'c

0

where the following governing parameters have been introduced:

L

(p.43)

(3.4)

Further Influence of E. and upon n can be

investi-gated by considering a continuous sequence of scalings, para-metrized with respect to a positive real vafiableoc in sudh:'

a way that for OC.-oO :

6=0cm

_A5)

=O(4)

where o and 0 represent the symbols of Landau notation,

E34} , [35] . In geometric termsrelations (3.15.) indicate

that as O. is made increasingly large the ship hull becomes

These parameters determine e.g. the similarity of

n upon S0 in the sense that for any two scaled problems

of the same and,18 parameters the distribution of ri

on S with respect to the nondimensional radius vector

(x,y,z) is the same:

OVLS

0J

13

-more and -more slender, with the breadth B and draught d

remaining of comparable size but increasingly smaller in comparison with the length L. Due to their geometric inter-pretrations the parameters and can be named respectively as the slenderness and thinness parameters.

The "time" vaiab1e t of the nondimensional problem is linked with the time variable t by the relation analogous to

(3.1.) and (3.2.):

; (3.4

with T representing the time scale. If it is assumed that the

spectrum of ship surface oscillations in the nondimensional problem .is accumulated about the harmonic of the period

I

the T scale represents the period of the harmonic about which the spectrum of the ship oscillations is assembled.

Employing the scales L,B,d and T introduced above it is possible to rewrite the equations (2.1.), (2.7.), (2.8.) and (2.9.) in the normalized form:

i1vt

D

-F

')-1. (")=O

,

ovt. 5,

(3.4s)

.',- 1

1o1.

where the following notation has been:introduced:

VU

(o)

-Besides in the above relations the asterisk..denotes

differen-tiation with respect tot1he nondimensionalvariable ,.

in the equations (3.19) and (:3.20.) several new nondimensional governing parameters have appeared in, .addition to the

para-meters

_{E. and18}

. The parameter denotes the reduced

represen-tative period of ship oscillation, defined. by one of the

relatiOns (3.21.). The parameter Sh, can be recognized as the Strouhal number of the problem. According to its definition

it can aisobe presented inLthe form:

with Fn denoting the Froude nUmber: '4

-Physically the Strouhal number .Sh, may be interpreted as the ratio of the periods of two oscillatory flow phenomena i.e.. the one due to the ship os'c-iila.tion with' respect to its referenca configuration and the other resulting from the passage of the.

ship as observed from _a fixed point in space, the period of

which is approximately equal to This interpretation shows

the importance of the 'S'h parameter for the interaction of the two flow phenomena. For the values of Sh being either .very large or very small, the two flow phenomena can be treated separately as slowly and fast varying with time res:pectiveiy or vice cersa. .if:the' two periodsare approximately equal Lc

the interaction constitutes n essential feature of the

flow and can riot be discarded.

From the definitionof V,(1.i)., it foiiows that it is

possible to express V in the fOrm:

with:

(3.2.3

5)

The interpretation of t as a governing parameter is obvious.

There is one more governing parameter, denoted by1, which is

implIcitly involved in. the equation (3.19). It is used for imposing the following requirement upon ship's surface

dis-placements with respect tothe reference configuration:

+ 8P i'

O ,

This condition implies that Is considered as a function

of as o(.- . It also follows from the equations (1.5.)

and (1.6.) and the condition (3.26), that.:

O(rL

(a

)

and;

'75'6

O(evi)

whereas:

(:329)

The relations (.3.28.) and (3.29) explain the notation of

the equation (1 .6.).

It has been discussed above how the relations (3.15) express assumptions concerning shipts geometry. The assumptions about

the ship motion kinematics are formulated in similar way by means of the relations:

.

-oo

The .assumption of similar nature concerning '' and

V will be discussed in the next paragraph.

According to the. physical interpretation of the Sh parameter

the assumption (3.30) means that the representative period

of oscillation of the ship about its reference configuration is

much smaller than theval.ue1, and it follows that the time.

dependence of the flow pehnomena,. resulting from the forward

5k

= 0 (4)

(3.Zo)

motion of the ship is negligible in comparison with the

dependence due to the oscillatory, motion. In view of the relations (3.27) 4 (3.30) the displacements of the ship surface with respect to the reference configuration is re-quired to be uniformly small on the surface with respect to the transverse dimensions of the ship, which, in fact is a necessary assumption for employing linear cylinder

oscil-lation solutions in further-considerations.

Referring back to the physical meaning of the Strouhal number Sh, it is worth mentioning that another interpretation of it

IS

possible in terms of frequencies. The frequency corresponding to the period T. is obviously:

(zi)

whereas this of the wave system movi-ij with the ship forward s-peed is determined by:

32)

It follows that 5h- can be written in the form:

5h--

(3.33) with: , -(.34-) 16

-Hence, in the sense of (3.33'.) the assumption

Sh.= o:('l'.

is

a "high frequency" assumption.

-The above considerations of the meaning of Sh.can be compa-red with the.order of magnitude assumption about U and

4. Simplified Equations.

In the preceeding paragraph a process. of scaling has been introduced for the governing equations (2.1.), (2.7 ), (2.8.).

and (2.9.), which produced the normalized governing, equa-tions (3.17,) (3.20..), and led to the definitions of the

governing parameters, (3.13), (3.16.), (.2.1 ) and (3.26 ),

of the problem.: Next,, several assumptions concerning the geometry of the ship hull, and the kinematics of ship motion?

were adopted in the form of relations determining the order

or magnitude of theparameters with respect to the scaling.

process, (3.15:), (3.30,). By means of these concepts the

governing equations càn be simplified in a systematic way. Beginning a.t geometric quantities, on the bas.is of the relations (3.10.), (3.11 ) and.(3.12), and the declared

boundedness .of the derivatives S0 S0 and

the normal vector n can be expressed in the form:

it = jC.) -with: (O (O1T2t/3)

= 1v1('t10o)

Cz) and:

or, as follows from (3.9-?): 17

-elM

'V

vs.

The expression (4.1-.) is an asymptotic expansion of the vector field ñ on S , in the sense that the firstk te±ms.

0 ..

of it approximate the components of. n with errors of o.()

18

-other quantities involved in the normalized equations (3.17 )

(3.20)

Without loosing generality, the equations (3.17.) (3.20)

can be considered as being defined, together with appropriate initial or radiation conditions, in the interval

(oi3

in the four dimensional space of the governing parameters

(,E.,Sh,,(') *)

For the procedure to be employed below be valid it is suff i-dent to assume that in a fluid domain 'a', which has as

0

one of the boundaries and a part SF of the adjacent free surface as another,. the following conditions are fulfilled:

all necessary partial derivatives ofLthe solution .with

respect to the "spacial"vairables are bounded in D at the point (1,1,1,1,) of the governing parameter space,

all these derivatives at any point of D are uniformly continuous:in the interval (0,1)Xhi of the parameter space.

The above assumption is adoptedherewithout proof.

It follows that in the space differential operator:

'(gL

-

Aa

-

'a?-''a"

-ar'

in the equations (3.17..) (3.20 .),can be treated as

a pure vector for the utorder of smallness" considerations,

with the order of smallness of its terms directly corres-ponding to the order of smallness.of the coefficient.s:.pr.e-ceeding the spacial derivatives.

Besides, according to the equations (1.10), ('3.21.) and (3.24), the time differential operator takes the norma-lized form;

( ( .

which is a um of the normalized time differential operator

in the x,y,z system of reference and a space differential

operator in the same system.The discussion of the time dependence.. of the flow pehnomena in 'the preOeeding para-. graph, indicates that the operator selects only phenomena due to the oscillatory motion of the ship with

respect to the reference configuration, and' owing to the It means that the values of each of the parameters are contained in the interval (0,1J.

choice of the T scale, it preserves the order of smallness of

the.

corresponding expressions. This can be written

formally as:

{)_= oc-n.

On the other hand the comments made above on the

operator

are fully applicable to the

space.

differential operator on

the right-hand side of (4.6c). in connection with this

the. importance of the order relations between the governing

parameters in the

scaling process

is shown. in (4.. 6)

by the presence of the quotient

.

These relations must

be subjected to assumptions similar to those imposed upon

the parametes themselves, in the sense that they are

necesaary

in order to obtain required proportions in the

representation of physical phenomena in the resulting mathematical model.

In the present considerations it is assumed that

.=

O()

as

oLoO

, which effectively suppresses the effects of

the drift speed of the ship in the lowest order of

smallness relations, as shown below.

On the ibasis of. theabove comments it is possible to rewrite the operators (4,.5) and. (4.6) in the form of

formal finitè.asymptotic expansions: '- . with:

and:

with:

/;t \"

I

\

-(b)'- S

19

-r4c (-,00)

H,

Ch): Ch)Z'

(4.9) (41o)

-. 20

-The relations (4.9.) - (4.12.) involve elements of the

notation already employed, in (4.1:) and (4.2 )' but applied here in a more general meaning. The numbers in the

super-scirpts indicate the formal others of smallness of the

terms, in such a way that:

(Q,

-ocir) 12.,3

f2)

OCiri1)

iL

4,2,3

where a.. denotes. an arbitrary expression and

lT,

signifies

one of the parameters

A sequence of order of smallness relations is assumed

con-cerning the parameters such that:

c,:

(

0j4))

Q

=0

Following the same convention, it can be found from (3.27 and (4.7 .) that:

.C4)

=

*(4) .

'.

r'1

(..'-

tiIuvi/

i.-.

-..--Similarly the equations (1.6) and the relations (.3.28 ),

(3.29) allow the tensor of rotation to be expressed as:

R (o)

+

C2 (3)

with R 3 and the subsequent terms corresponding to the terms on the right-hand side of (1. 6 ).

With the remaining terms in the equations (3.17) (3.20)

interpreted in the same way, see Appendix C, the solution: (4A

I

can be sought in the form of an asymptotic expansion:

: &I) "L2.)

in which the notation defined by (4.13 ) and (4.14 ) is

applied..

Some further details having been explained in the Appendix

C, it is found that:

o

(I.4

since the lowest order of,.smallness, independent terms in the equation (3.19) are of Q(ø.9') and the assumption'..'

(4.19) is also compa.tiblë' with theoher equations. CL)

The equatiorfulfilled by and

,in D and

on

can be sunmarized as follows:

(}L)tcc3))o

_I with: (t

sM1

1- _CE

tLe' +

and:

Sh i+

F'14t L:i.±*

1 -+

___

___

I,-

-az

i-

'.

I Opi S .

-

21

-

22

-It is worth noticing that the bouhdary condition (4.21.) for the potential 2)does not involve any terms expressing coupling between forward speed and ship oscillation effects-Whereas such terms appear in the boundary condition (4.22 for the ptotential Two such terms, which are underlined in the equation (4.22.), are present to various extent in strip theories, e.g.

_[131,

El4 ,

[151.,[9]

,

.[ioj

,

cii]

, whereas.

those implicitly involved in the derivatives of the poten-tial have usually, been neglected, compare with a

conunent in [2'lj

In order to establish the free surface boundary conditions

for the potentials and. , the order of

small-ness of the. governing p'rameter T. . mus.t be chosen. As

men-tioned before, such a choice should be based on the. rela-tive importance of the physical phenomena under

considera-tion.

The comparison of equations (3.18), (3.20.,,) and (4.6) reveals that wave making effects due to the flow pheno-mena described by the potential wil1 not be suppressed

in the mathematical model by the smallness of the slender-ness parameter . ', if it is' assumed that:

= O(

The insertion of the relevant expressions into the equation (3.2O), taking into account equation (2..15;,,),..ields the following asymptotic expansion for the wave elevation an

'- (t')

-F...

where:

f9_.\

-)

-

- s.

+

/

.;M

)'

4

's']

-,

and

i41f,-

t_

The equations (4.20), (4.21i, (4.22.), (4.26.) and (4.27 represent a simplified set of governing equations which

corresponds to the basic problem..defined by the equations

(3.17) 4 (3.20 ), and they must be supplemented by

appropriate initial or radiation conditions. In other words if supplemented .in such a.waythese equations define boundary problems from which the terms and C3)f the expansion

(4.18.,) can be found.

However, since the governing parameters 6Sk1'7, may

change arbitrarily within the limits imposed by the con-dition (4.14.,) the potentials and are not

homo-genous in their orders of smallness,.and as can be deduced

from the equations (4.21), (4.22) and (4.27 ) they should

be considered in the respective forms:

-

(M)

Y(Y1)

-

23

-with the expressions on the right-hand sides of (4.2Sj

to be evaluated at '= 0.

From the equations (3.18.), (4.2:4f,) and (4.25.) it follows that the free surface boundary conditions for the potentials

and on SF can be put in the form:

E

'az.

Obl. 1=0,

(4.24)

and:

[

:. -1.(a..\ i')

_{- ZSk(}

L\

za1

£

J4'L

'

_'a + C2.)

-O4O.

1-

cv,yt

(5)

(4.tS)

e.)

The subscripts in the expressions on the right-hand sides. of (4.28. ) and (4.29.)) indicatc precisely the order of smallness, of the terms they are attached to, according to the notation defined by (4.13 )'. Following strict links between the governing parametes and flow phenomena the.

subscripts give also the physical int'erpration of the terms. By insertingthe expressions (4. 28.) and (4. 29.) into the equations (4.2O.), (4.21:), (4.22,), (4.26;) and (.4.27. )

and equating terms of the same order of smallness, the

governing equations for the six potentials

can be found. These are presented.in the Appendi C.

However for c.omparisonwith the results published in the. literature and for some further considerations it is con-venient to express the governing equations in dimensional form which can be easily obtained by means of the relations

(3.21,:) sz.(; .'

'(shE'(v&))0 ,

D

j-05h0

OYI.

2.0

i,,.. - 24'. oyl. -I49') J3 OVL S0

and br '.t!e thir.d order :P0te,15:

b) + 1

o

z,. = 0 i..n D c.). [

*1

,opv7_'O1-r.D d) _£ f',-s1. 'p a.) b). c.) e )

+ v11Sli)

VJ')1

on S

(rt,4

1-

h

= L(v5W3 .Z& z:)i-

--

.p N3 rL5 x I OI'

(j5

-

'JL}Z)

-

t(v)1%?},x + + - 1sX

1i4'5' (Nt

I

in the above

equatiorO

denotes the image of D according

to the transforation(3..2).. It should be pointed out.

that in general:

t.

The reason for this is roughly that the scaling (3.2..) looses

the.

requireddproperties (see the assumption a), b.) on page 18) far enough from S in .D.

0

In the part of D where (3.2J is not applicable the appro-priate scaling should be based upon wave dimensions rather

than ship dimensions and the. corresponding solution for

velocity potential should be matched with the near field

potentials etc. providing, appropriate patterns for the, behaviour of the near field potentials for from

se'. ). In particular. such a process provides necessary

in-formation about the dependence. of the. solution upon the

X variable, which is lacking in the equations (4.30.) and (4.. 3'!1 ). as. the potentials in them' can be added to or

mul-tIplied by an arbitrary function of X. Such processes of matching have been described e.. in E16] , [171, E18] , 2i

f22] , [231 1and t24]

However, in strip theory potential f lows induced by plane. D comprises the neighbourhood of S in which y, z = 0(1) and shOuld matter the far field solution for' y =

O(fl.

-

25.

26

-oscillations of cylinderin domains unbounded apart from

the cylinder contour, free suface and possibly sea bottom., have been employed in order to represent the flow induced by the oscillating ship, see e.g. [13) , t14] ,[9], tIO3:or t11

This kind of' approach can be generalized for the use in the present considerations by the assumption.:

=

Taking into account possible differences in approach to the matching problem, notation and using of the components of

instead of N, the equations '(4.30:) and (4.3i,)

can be compared with analogous relations reported in the

literature.

it is found that the equations determining potential

((4.3O. ) a), b), d)) are identical with the corresponding

equations derived in [16], [211. and 1221. For P15

the equations (4.30.) a) c.) e) which determine the poten-tial are the same as. the corresponding equations reported in [29] . However the. equation (4.30..,: e) for

is different from the corresponding condition (5.2Oy'.) in [161 , as it does not 'involve the term

which according. to.the present derivation is of

O(a.C)

in the. normalized form..

The same difference occurs in comparison with [22]

The potentials and shoul6 be considered

jointly for comparison with f22) (for 0 ) and

ç20] (for

0).

Apart from the difference mentioned

above in connection with the corresponding versions

of ëuations (4..30) a), b), c) and (4.3i,) a),c)',f) added together are found to be identical With the respectiv&

equations in [221 and [20] . Besides the equations .(4.3P)

a), c), f)for1v=O are the same as the corresponding

equations 'in t161

Finally the equations (4.31.,i) a.), d), g) determining the potential coincide with the. analogous euqations

de-rived in t29] for

In addition to the. properties discussed 'above, similarity features of the equation's '(4.3'0.) and (4.31,.) should be pointed out. From the method by which these equations

correspond to the same.:

normalized reference hull surface S governing parameters: _,8 1Sk, '

nondimensional

state variables:

- -..

-.

j.

_-' % * * - N.

and motion history.

The dependence on the motion history will. be indicated later.

Experience with ship motion pr-edictior'hosthat usually

the requirement a) can be-considerably relaxed allowing

for geometric similarity to be basedon the identity of some general form parameters, see e.g.. r273 ,C3-7]

Finally it should be mentioned that for the consideration of ship motions control it is sOmetimes more convenient to express the effects of the surrounding flow in terms of quantities observed in t:he reference system fixed in the ship. The, corresponding transformation is presented in the Appendix D.

5. Memory ffects..

The equations (4.30) and (4.31L) involve several boundary

problems which if parametrized with respect to can be

written in the form,:

.i +(L'1O

_'btbJ

-

27

I_vt.

t:)1.:'(

OVL- Z.

0, L.1f, 0

ak

(SA)

In the equations ('5,1,) C0(X denotes the contour of the cross-section of S0 at .X ( i-s assumed to be identically

zero if the set- of points on S at .X is empty or

reduces to one point) Besides from the definition of,see (4.4 )) it follows that, N2 and N3 denote the components of the normal to C0(x) at the' current point Q on -th&con-tour. A-s an example the equation- (4.3O:. a) can be expressed as:

28

-(w4, i. _i'13ij

'(vL&'

c1(Q')

(5. 2.)

with:

which allows to present as the sum of the solutions

of three problemsoOf the type (5.1 ).

The potentials

_(k)

LY1EY ,

(tsare

also determined

by this class of problems which are reducible to the form

(5.1.), as well as the'potentials and

where:

E

(v1ZE.)Z.

and ' fulfil the homogeneous

con-dition on z=O combined with nonhomogeneous concon-ditions on C0(x), whereas corresponding nonhomogeneous conditions are

applied to on z=O combined with the

homogeneous condition on C0(x).

In all problems of the kind characterized by. the set of equations (5.1,) the condition 'b) provides for the pre-sence of so called "memor.y.effects" which correspond.to the fact that if . r(t) is put equal to zero after some

period of time when (t). 0, the waves created due to

the previous motion of the boundary persist for the inf

i-nite period of time in theideal fluid. These effects were

first considered in {38]., and a comprehensive discussion

can be found in f12J

The original idea of was to express the

_Iuionof

a problem with the free surface condition (5.1 b) as a

sum of an impulsive and memory effect terms. This can be

carried out by writing:

(g.3)

and:

t-(b)q ftr(&)CO)

--

(f.)(FCP(0)

('t

whereby the condition (5.1 b)can be put into the form:

S1A)Ct)U +

-1)ctt i-

3q,

4-i-('(0)]+ t(t(f(O') t

L1'.O, o'i.

zi.O.

Taking into account the physical interpretation of the

potentials _4' and and the form of the free

sur-face condition (5. 9 ), the potentials can be charac-terized by the following boundary problem:

a)(1

_{'- )(iqc6=o}

vfl.

Di

0.b .x

'.'O

b)(Wzi..

+3

(,

4f)((q10),

C0(,c'

c)[9h+ (I

(q(.'O

or zO , tvD

d)qi=O

It follows that: + 29

-where &(&) denotes the Dirac's delta

function.

Hence

t) represents the impulsive paof whereas (-')

cha-racterises the memory effect. The equality (5.5) can also

be rewritten in the form:

and the initial conditions:

(p(4)=O

0n Z.011,vkD,

fot0,

p(4_

jL

on. zO

30

-The initial conditions (5.10) are less restrictive than

those proposed in where the condition corresponding

to (5.ai) a) requires in _D for 4=0.

It should be pointed out that the number of the problems

characterized by the equations (5.10) and (5.1i), which

should be solved in.connection, with the problems (4.30 )

and (4.3] ) can be greatly reduced if the solutions of.

some of them are used for approximating. the solutiors of

others. This can be accomplished by representing the

function of a given problem in the form of:

__(Q=

46(Q'

(5.41)

with.(Q) denoting the

functions of the approximating problems (e.g. _(Q)1 1=2,3,4, as defined by (5.3 )), and

being the solutions of the normal equations:

b

4(Q

cLC(Q

f0'.

c42

It should be noticed that if the(Q) defined in (5.3)

are employed the lowest order potential in (4.30. )j is to be solved for in the exact form..

Further explanation of such an approximation and examples

31

-6. Conclusion.';

In the present report a 'set of simplified boundary pro-blems (equations (4.30.), (4.31.), (4.33'.), (5.10)) and

(5.I)') has been derived, which characterizes in the time

domain potential flow phenomena induced by the ship moving with a steady velocity and oscillating as a rigid body

about the reference configuration.

The derivation was based upon the general boundary

pro-blem (eq (2.1), '(2.7), (2.8') and (2.9))which by

means of a scaling' procedure (eq., (3.2) and (3.16 .))has been reduced to the normalized problem (eqc (3.17')

(3.20)). The essential physical assumptions have been in-troduced by imposing appropriate, order of smallness

re-lations upon the governing parameters of the nondimensional problem (eq..i (3.131, (3.15 ), (3.16.), (3.21.), (3.25 )

-(3.30), (4.81, (4.13), (4.i4).and (4.23 )). Such a

procedure is different from. that usually employed in slen-der body slen-derivations, see e..g. r16] and r18J, as it intro-duces as many independent small parameters (in the present

case ) as there' are fundamental approximating

assumptions. Besides it puts emphasis. on the similarity

fea-tures of the problem which are of fundamental importance

in applications to design £27]

Due to this approach', in the present derivation:

the role of the order of smallness. of' the forward

speed U hascbeen elucidated by the introduction of the small parameter Sh (S.trouhal number, eqc., (3.22:)) and the meaning of the "high frequency" assumption has been clarified '(eq( 3.33)).

'the order of smallness of ship displacements from the

reference configuration has been explicitly determined

as uniformly small on the hull surface (ec.(3.26.).). The

assumption which so far has been tacitly adopted by

:'.érnploying hydrodynamic forces induced on cylinders

oscil-lating with small amplitudes. As a result the pitch and

yaw angular displacements 'have been found to be of higher

32

-displacements (eq.:;. (3.27.) (3.29)). This partly accounts for the elimination of a higher order term

(zv

) from the boundary condition of the final

problem, in comparison sith the results presented in _£16] and _[22]

c) it has been shown that unless the basic small

para-meters ( ) are considered as being of different

orders of smallness (with the assumptions (4.13) and (4.14) fulfilled), terms nonlinear with respect to

the ship oscillatory displacements appear in the con-stistent third order formulation of the problem,

to-gether with the forward speed effects. The nonlinear effects have been found to be compatible with those

derived.foran oscillating cylinder in [29]

Referring to time domain and frequency domain approaches,

by simplifying the boundary problem in the time domain-in

the present derivation, the memory effects have been

re-duced to two-dimensional phenomena, which have already

been successfully handled, e.g. r4oJ

Besides some further simplifying procedu.reshave been

pro-posed (eq.. (5.12.), (5.13 )) by means of which efforts required for the solution of the problem can be reduced.

Taking into account the importance of the influence the f

foward speed and nonlinear effects can have upon the

assess-ment of the stability of the ship motion and the effective-ness of stabilizing devices, it is. rcommended that an

attempt should be made to compile an extended strip theory

algorithn that would be consistent with the derivation of the equations (4.30) and (4.3 ).

Appendix A.

A Comment on the Criticism of the Superposition of Potential and Viscous Effects in a Fluid Flow as

pre-sented in

_r6]

In this comment the set of equations describing a two

di-mensional viscous flow with a free surface, as presented in _E63 , is discussed. These equations are written in

terms of space variables normalized with respect to half

B .

beam of the ship

j)and time variab1enorma1izedwithrespect to

(A) which denotes the angular frequency of the oscillation.

Besides the nondimensional numbers Rn and Fn are defined

as follows

The flow velocity field V is represented in the form:

where:

(6)

with and ebeing versors along the x and y.axes

res-pectively, and the original enumeration of the equations in t6] is indicated by. numbers not labled with A.

According to E6 the functions and should fulfil

the foliwoing set of equations:

i- 4

in the fluid domain,

33

34

-on the surface of.the body, where Uand

_Vbdenote

appro-pria:tly;f normalized velocity components on the body sur-face, and:

+(--)o

C1rt 13)

on the free surface.

The condition (10) can be expressed in the form:

(z)

where Wdenotes- the normal to the body surface and _Vn the

normal component of the body surface velocity, and:

with and representing the components of the tangent versor at the body surface.

The 4bove set of equations can be expressed in the form

of two coupled boundary problems:

in the fluid domain ,

)

on the surface of the body,

-2L

_)Ofl

-..

the free surface and:

35

-(viJ- v=O

(v1

on the body surface

bt

c2

(t_O

_{on the free surface}

Taking into account that the term on the right hand side

of(3.A c) can be dropped for sufficiently small - ,

the problem (3A) supplemented with the appropriate raditation

condition presents the classicaLpotential flow formulation for an:rioscillating body.

It can be solved quite independently from the problem (4A)

and next the solution can be used for the evaluation of

the exciting terms in the boundary conditions(4A b) and

('l.A c).

It seems necessary to check if the problem (4.A) is well

posed. The negatve answer would uggest that the form (6)

of the velocity field 'is not appropriate.

Appendix B.

The .erivation of the Rotation Tensor R.

Let and e. i=1,2,3, denote respectively the versors

of the initial and final position of a rotating system of right-handed áartesian coordinates. The rotation tensor

corresponding to these two positions of the system is defined by the equation:

/ for 1=1,2,3,

(1.B

where the dot denotes scalar multiplication.

From (1B) it is clear that can be written in the form:

=

3

j1 J

J

with ø denoting dyadic product, or by means of the

re-presentation in the

,

1=1,2,3, system, as:

with:

3

Assuming that represent the rotation angles in

radians about the axis x' ,y" and z' respectively, takeh

intthé indicated order, the final rotation tensor, can

be written as:

R5&R14)

(s.

or as a representation in the x,y,z system:

j-

RkR$kR.

k1L

From trigonometric relations an6 Taylor series develop

=

4

₀

o

CO94

o

o 00\

₂₁₀₀₀

o

0 -1

)

(o

1

0

ci 0/

00..i

it ₃

0 00 \

0

0

'1 ₎ -f

0

-1

0 1.

/ Co'YLs

0

tTh115

\t

0

4

0

0

1

004

a

ci o

-1

i

0

c.o.

0

(0 -4 0

0

₀

0

00

-

3.7

-0

0

4

/ '

00

O40

\o 01

/i oO'\

0

.1

0

'.

0

Q4 I

L

° \

000 \to(

_00/1/

(i oo\

(o

o

0

'1/

It (

04 0

\

00)

38

-By inserting (7.B) (9.D) into the equation (6B),with

taking into account the order of smallness relations (3.28 ), (4.13 ) and (4.14 ) the equation (1.6 ) can be

Appendix C.

Normalized Simplified Equations.

The derivation of equations (4.21 ), (4.22J, (4.26,) and

(4.27 .,) requires that the normalized relations (3.18 ) ,(3.19,)

and (3.20.) should be used in the formcorresponding to the

equations (2.14), (2.15) and (2.16 ) respectively. Taking

into account the definitions '(3.21 ) the nondimensional

forms of the last three equations are found to be as

fol-lows:

,rc)2.

2 ,c

[t(3

o('}

o-

=0, ) - 3,9 -1 I.

R ;

/ .

+ C R

c')

' - .

(c')

t. I and: *

Sh (4,

&(

1-In addition to the relations (4.9,), (4.10.), (4.11.), (4.12. I

(4.15.) and (4.16 ) in developing the equations (1C) (2)

with: and: with: ( (

C-vet3'

40 -(é.C')

+Ls

4.

(-,7)

=&*bit

%*

..,

$

*

.5t'_PiLl4L I_z

+

'-'L -t.i'.)

':i CSh.) C

pL

Skf\11 ,

oi-0

By inserting appropriate expressions into the equations (3.17 ), (1), (2.C) and (31), and by taking into account the developments (4.18,) the equations (4.19.), (4.21 ),

(4.22.), (4.25.:), (4.2.:) and (4.27,.) are derived.

Next the emi'oymentof the developments (4.28..:) and (.4.29.) yields the following set of normalized governing equations:

..qI * _/ e)

(NJ-.

+

-*

N3, 5.0 S.c)

g)

41

-and for the third order pOtentials:

J

'cYLZ)I

,v%

I,

+

L'r:i

_1L11E2

0

,

0

[6s

₊

c&

.)[

+

_U\

'I _LMD [

aa( JfrLEicar

_+4.Ft

c _(EP oltO, -I. '(r Sti) Ce N11 Sk O'(V)

(3()

i -

4)

-

NI3)+

*

lJ3)])

øi S0,

2'"<

J3*

W + .+

which correspond directly to the equations (4.30. ) and

with:

Taking into account the definition of R1, see eq'(4B), (].D) can be rewritten as:

42

-Appendix D.

Transformation of Kinematic Quantities from the Inettiai System of Reference into the System Fixed with the Ship.

In this report the motion of the ship and all resulting

kinematic quantites have been expressed in terms of

components in the inertial system of reference x,y,z. However for' the investigation of.ship stability it may

be more convenient to use expressions referred to the system of reference fixed with the ship, whichcoincides

with the x,y,z system in the reference configuration. Below components in the .fld in the. ship system will be labeled by primes. Since only rigid body motions of the ship are considered in this system the material points of the ship' preserve their reference configuration co

ordinates. Besides,among the quantities charcterizing

the ship motion are considered by

definition to be independent from the choice of the

sytem of reference,.

It follows from (3.19 ) that the relation of interest

is of the form:

In order to match the transformation ith the asymptotic expansions' applied in the report it is necessary to find:

*

(4)

43

-From (3.D) it is found that:

(si,, (5' 2.

(S

R. (o)

(s+,)r1=

Ci

+

£

-'

3)

* '4)

C)

(5ht3-r).

=(+'

R +

4

/ Ct) Cfl ' C3) £O

t(s&

+-E;t-J3

S

It follows from(5.D a) that

(ioo

'C)

(ooO

With (6.D) inserted into(D b) it is found that:

I,

(oo,o)

/

(4)

-! (i

..

(A\

_)l)VLa.1Y73

/ ,J -?

In the same way from c):

Summing up: and: -I (2.) L'

.(Ou,O)

-C1)

rt(2)

'L

't

_tSl5Ld4ct

*

*

/1.

(...J '1

(l

1 '.. Y111 ₊

&

_V14%113

jt

5+V149()+

')

(D)

-p

0

fl1 2.13,

(Q.D) a) P (s.o)

44'

-The relations (5p) and (1O.D) are sufficient for

transforming relations up to the third order of small-ness.

4,5

-References.

[i] Geoffrey G. Cox, Adrian R. Lloyd,

"Hydrodynamic Design Basis for Navy Ship.Motion Stabilization", Trans. SNAME, Vol. 85, 1977 pp.. 51-93

t2) Rodney T. Schmitke,

"Ship Sway, Roll., and Yaw Motions inObligue Seas" presented at the Annual Meeting of SNAME, November

16-18, 1978.

[3) Yoshiko Ikeda, Yoji Himeno, Norio Tanaka,

"A Prediction Method for Ship Roll Damping", Report of Department of Naval Architecture,

University of Osaka, Preferture, No. 00405, December 1978.

t4J St. Denis, M., Pierson, W..J.,

"On the Motions of Ships in Confused Seas", Trans. SNAME, Vol. 61,. 1953, pp. 280-357.

W.G. Price, R.E.D. Bishcp,.

"Probabilistic Theory of Ship. Dynamics", Chapman and Hall, London, 1974.

A. YUcel Odabasi,

"Hydrodynamic Reaction to Large Amplitude

Rolling Motion",

BSPA, Wailsend Research Station.

Bispiizghoff, R.L., Ashley, H.,

"Principles of Aero-elasticity", Chapter 10.

[8) R..E.D. Bishop, M. de A.S. Neves, W.G. Price,

"On the Dynamics of Ship. Stability",

Nils Salvesen, E..O. Tuck, Odd Faitinsen, "Ship Motions and Sea Loads", Trans. SNAME, vol. 78, 1970.

io1

Alfred I. Raff,

"Program Scores - Ship Structural Response in Waves", U.S. Coast Gard Headquarters, Washington

D.C. 1972.

[ii] R. Wahab, J.H. Vink.,

"Wave Induced Motions and Loads on Ships in Oblique Waves'; Netherlands, Ship Research Centre TNO, Report No. 1935.

T. Francis,Ogilvie,

!IRecent Progress Toward the Understanding and Prediction of Ship Motions"

Fifth Symposium on Naval Hydrodynamics, September 10-12, 1964. Bergen, Norway.

B.V. Korvin-Kroukovsky., W.R. Jacobs, Pitching and Heaving Motions of a Ship

in Regular.Waves", Trans.. SNAME, Vol. 65, 1957, pp. 530 -. 6:32.

[14) J. Gerritsma, W. Beukelman,

"Analysis. of the Modified Strip Theory for Calculation of. Ship Motions and Wave Bending Moments", international Shipbuilding Progress, Vol. 14, August 1967, No. 156, pp. 319 - 336.

t15J Michael S.. Triantafyilou,

"Strip theory of Ship Motions in the Presence

of a Current", Journalof Ship Research, Vol. 24,

No. 1, March 1980, pp. 40 - 44.

[161 T Francis Ogilvie,

"Singular-Perturbation Problems in Ship Hydro-dynamics" Adv. Appi. Mech. 17, pp. .81-188.

47

-T. Francis Ogilvie., Ernest 0. Tuck,

"A Rational Strip Theory of Ship Motions: Part I" Rep. No. 013, 1968, Dep. Nay. Archit. Mar. Eng., University of Michigan, Ann Arbor.

l8] J.N. Newman,

"The Theory of Ship Motions", Advances in Applied Mechanics", Vol. 18, 1978, Academic Press, Inc.

[19] Armin Walter Troesh,.

"Sway, Roll and Yaw Motion Coefficients Based on a Forward-Speed Slender Body Theory - Part I','

Journal of Ship Research, Vol. 25, No. 1, March 1981, pp. 8 - 15.

r2o] Armin Walter Troesh,

"Sway, Roll and Yaw Motion Coefficients Based on a Forward-Speed Slender-Body Theory - Part 2".

Journal of Ship Research, Vol. 25, No. 1, March 1981, pp. 16 - 20.

J. Nicholas Newman1 Paul Sclavounas, !IThe Unified Theory of Ship Motions",

13th Symposium on Naval Hydrodnamics, Tokyo , 1980.

Hiroyuki Adachi, Shigeo Ohmatsu,

"Analysis of Ship Motion by Slender Body Theory", Reports of Ship Research Inst. Vol. 14, No. 6, Ship Res. Inst. Ministry of Transport, Tokyo, In Japanese.

Hiroyuki Adachi, Shig.eo 0hmatsu,,

"Analysis of Ship Motion by Slender Body Theory (Part 2) ",

Reports of Ship Research Inst., Vol. 15, No. 3, May 1978, Ship Res. Inst., Ministry of Transport,

48

-124.] Hiroyuki Adachi,

"Analysis of Ship Motion by Slender Body Theory Part 3",

Report of Ship Research Institute, Vol. 17, No. 3, May 1980, Ship Research Inst.,Miñistry of Transport, Tokyo, in Japanese.

t253 S.J. Kline,

Similitude and Approximation Theory", Mc. Gr.aw-Hili., New York, N.Y. 1966.

t26] Lee A. Segel,

"Simplification and Scaling"

SIAM Review, Vol. 14, No. 4, October 1972, pp. 547 - 571.

t273 Jacek S. Pawlowski,

"On the Application of Non-Structural Models to Ship Design",

to be published.

[28). J.N. Newman,

"A Linearized Theory for the Motion of a Thin Ship in Regular Waves",

Journal of Ship Research, June 1961, pp.. 34. - 55.

E29] Roger L.. Potash,

"Second Order Theory of Oscillating Cylinders", Journal of Ship Research, December 1971,

pp. 29.5 - 324.

[30J H. Sôding,

"Second Order Forces on Oscillating Cylinders in Waves", Schiffstechnik Bd. 2.3 - 1976, pp. 205 - 208.

t31) A. Papanikolaou, H. Nowacki,

"Second Order Theory.of Oscillating Cylindersin a

Regular Steq Wave",

49

-Jacek S. Pawiow:ski,

"The Impermeability Condition on the Surface of a Deformable Body",

Unpublished note.

J.J. Stoker, "Water Waves",

Interscience Publishers Inc., New York, 1957.

Jean Dieudonné,

"Infinitesimal Calculus",

Kershow Publishing Company Ltd., London 1973.

E.T. Copson,

"Asymptotic Expansions",

Cambridge University Press, 1965.

P.A. Lagerstrom, R.G. Costen,

"Basic Concept Underlying Singular Perturbation Techniques",

SIAM Review, Vol. 14 No. 1, January1972,

pp. 63 - 120.

Jacek S. Pawiowski, K. Kwiek,

"The Computation of Ship.Motions in Longitudinal

Waves, Based upon the Main Particulars of the

Underwater Form" IVth Symposium of Ship Hydro-dynamics, Gdansk 1977, in Polish.

[38] W.E. Cuniinins,

"The Impulse Response Function and Ship Motions",, Navy Department David Taylor Model Basin,

prepared for Symposium on Ship Theory. Hamburg, Germany, January 1962.

L39] Jacek S. Pawlowski,

"The Estimation of the Diffraction Force Components from the Equivalent Motion Concept".

50

-t401

Hiroyuki Ad'achi., Shigeo Ohinatsu,,

"On the Time Dependent Potential and its Application to Wave Problems",