Basic relations of strip theory. Part II: Elements of the equations of motion

LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

BASIC RELATIONS OF STRIP THEORY Part II

ELEMENTS OF THE EQUATIONS OF MOTION

Jacek S. Pawiowski

Report no. 558

August 1982

Deift University of Technology

Ship Hydromechanics Laboratory Mekelweg 2

2628CD DELFT

The Netherlands Phone.015:-786882

Contents.

page:

Introduction.

Basic Relations. 2

The Radiation and Hydrostatic Forces.

The. Equations of Motion. 16

Conclusions. 25

References. 27

Appendix A. The Estimation of Integrals over

the Image of the Wetted Surface. . 29

Appendix B. Asymptotic Expansions of the

Cross-sectional Hydrodynamic Force and Moment. 40

Appendix C. The Weight and Inertia Forces. 56

Appendix D. Memory Effects and Classifications of

Potentials.. 63

Appendix E.. Potentials of Flow Velocity for

Harmonic Oscillations.. 73

Appendix F. The Frequency Domain Representation

of Cross-sectional Forces. . 7,9

Appendix G. .The Equations of Motion in the

The present report constitutes a continuation of Part

in which the schemes for the integration of. pressures over the hull surface and derivation of those pressures, to a formally arbitrary level of approximation,, compatible with the considerations of Part I are established. They are pre-sented in the Appendices A and B. According to the schemes the expressions for cross-sectional radiation and restoring hydrodynamic forces up to are derived in.the

Appendix B and the same approach. is extended to cover mass

and inertia forces in theAppendix C.

Since the derivations have been carried Out in

the time domain,further discussion of memory effects, aiming at a proper classification of pressures and forces, and the preparation of a passage to the frequency domain., are con-tained in the Appendix D.. The links between time domain and frequency domain potentials for steady oscillations are discussed in the Appendix D, and are followed by the presen-tation of. formal expressions for the cross-sectional forces in 'the frequency domain, in the Appendix F. The presentation of the effective means for the evaluation of these forces

is out of. the scope of the present report.

In the main part of the report the results obtained in the Appendices are discussed in the context of an enlarged strip

theory involving essential nonlinear effects.

"Basic Relations of Strip Theory., Part I," Report nr. 544,

By definition the total .hydrodynamic force exerted.on the ship by the ideal fluid, and the total hydrodynamic moment

ti with respect to the point P fixed with the ship, are

expressed correspondingly by the formulae:

SW

p(---f)'%c&,

with

5,

denoting the wetted surface of the. hull. It is convenient to introduce the following normalized quantities: 'BL r

rTt

I °

?_ 1')'T'

d

dL

In terms of them the equations (1.1.) and (1.2.) can be rewritten as:

1

1=-s 3-'

For the case of stea4y motion of the ship the following definition is adopted:

T\fF

which preserves the order of smallness relation (1.4.22.)

Numbers of relations which are preceded by I refer to relations in Part I.

and the amplitudes of oscillations are assumed to equal zero.

It is assumed that at rest the ship remainsin equilibrium under the action of the force of gravity. This can be expressed as:

with

S(Q

denoting the normalized wetted surface in the equilibrium configuration, and:

(4M

with and d signifying the specific mass and volumetric element respectively, so that the following relation holds for

the mass element of the ship

cki.,:

(i.9)

Usually the equilibrium configuration, which is assumed here to be uniquely determind by the conditions (1.7.),.is

identiefiedwith the reference configuration for the

compu-tation of varying in time hydrodynamic reactions resulting from ship oscillations. Here however, the reference configu-ration is considered to be. the one in which all time independent forces acting upon the ship remain in equilibrium.

The conditions of the equilibrium can be. put in the form:

S

C-

StT-i

_{() +}

_E-

(S7(

2

*[

₍₅

+

-where 50(.L)de.notesthe wetted surface of the hull in the

reference configuration for the steady veloctiy (due to the time independent forces being in equilibrium.) regime of motion.

enotes.the velocity potential corresponding to the boundary problem (1.3.17.) - (1.3.20.) with the following conditions imposed.

Y=O)

O

$E and in (1.10.) represent respectively

+1enor-malized steady force and. moment., and include in.particular vis-cous and propulsive forces as well as average forces due to ship oscillations,, wave excitation etc.

In order to take into account the equilibrium of forces in the. reference configuration, the expressions (1.4.). and (1.5.) are rewritten in the form of integrals taken over the reference hull surface 3, :

tç=.

(

d),

OVJ with:

i.e. is the inverse image of

S

according to the map-ping (1.1.2.).

The ingetrals over S0in (1.12.) can be expressed as the

where:

and:

Oj

oJuu1

&SOLAJ C&l.1 ,i( - SQ%.AJ '1

The pressures in (1.12.) can. be classified, as those applied on in the absence of. oscillation (the subscript S will be assigned to them) and those due to the oscillation of the ship

(with the subscrL±ptD). Then the same classification follows for the force and moment I'1., yielding:

and:

t,=.s

t -_\

.(''-i-

d.

i

-t.C.

._]

+

LEj

-5

1AI

jv)

Uk'

P-@15)

and I1p. are)according to the adopted definitions,represented.

in. the equations (1.10.) bythe explicit expressions of

hydro-dynamic forces. . This is due to..the fact that the influence of

the drift velocity is suppressed in and t1p up to that cc:.

order of smallness. Once the reference configuration is known

these.forces are of little interestfor the determination of

ship oscillations.. Hence further consideration are concerned only with and FI

2. The Radiation and Hydrostatic Forces.

In the Appendices A, B and F it has been shown that the gene-ralized components of the radiation and hydrostatic restoring forces, defined as:

1k -

(D M11;)k

p k =1,1....

(2. 12)

can be expressed in the form:

4

fo

k=

6,

with denoting the coordinates of the extreme cross-sections. at the stern and bow respectively. The cros-sectional

for

k

Ii L,.... ', and L.

0,

. where the indeces .- ,j

re-fer to the cross-sectional modes of motion that result from the perturbation analysis. The expressions for and in

the time and frequency domains (for steady, oscillations 'fin the latter case ) are presented in the Appendix F, for the ship with lateral symmetry.

By substituting (2.3.) in (2.2.) the components can be found in the form: I' ' LI

fk--

t

for and

4

F

_1k1tX)

cL'.

where: (z. .5) forces

R.ic are determined by the formulae:

RK

o(

a43

ir15

1S - 643, _/

k

for k4,z..., and j,.Zj , in which the linear part

of the forces is comprised in the first sum on the right-hand

side.

It follows from the formulae (6.D.). (14.D.), (17.D.) (24.D.) and (33.D.). that the cross-sectional modes of motion 2 and3 split into the modes 2 and 6, and 3 and 5, correspondingly, when ex-pressed in terms of the ship rigid body modes. Hence the com-ponents can be rewritten also in the form:

+ 2

-t

( o)

where the index t. in the first sum on the right-hand side in-dicates shipts rigid body modes and the prime is used in order to indicate that the indeces of the second sum denote the cross-sectional modes.

In the frequencydomain.the linearparts of the components

can be expressed in terms Of added mass, damping and restoring force coefficients, denoted by A

'k

and C _kL. respec-tively:

2:

A KO

(-wA

t +

L1'

It follows from the equations (36.F.) 4 (43.F.) and (2.5.) that:

ptt- .k

-

_Z2 - _{2. I}

\

1-1Z.O4

=

&2.oI

''LG

Jxaa.z+

XtL21 R(S&+S'ao)

24 i

Lti0

q_33-l-

_{2J 303}

1333- $b33

_/

C33b,

+

_1-b3

33 2..I..

b35

-x'b3+

o_ 5641..

-=

-- St'

c#4s1=

9(Th1

c'Q

c4

-A51

S'aai

151

A5_$Q3+

_S'603,

k5=

ç(x'-a 1-3

655

Lt S(I)2..Q

C55

x97

I

- 2

(

x'

+

-

(A(Qi-rjuc'a20,

LOZ. ,

'2=

S

+ Lt (

o,_+

+ LL _/

stood to be carried out with respect to d.. from to

, C denotes the local breadth at the waterline,

&1,-i- the transverse metacentric hight , the coordinate of

the centre of buoyancy and A is the local cross-sectional

area.

Besides the following definitions are adopted:

--

+

-

o

-bkOL

-

+ Lz2 b'kOti,2

+

*

for

k=t

,

k=3

, W=

Z ,

and C. 4 , with the terms on

the right-hand sides defined by (34.F.) and (35.F.). In (2.10.) the terms involving Dirac's. delta "functions" have been in-cluded formally for purposes of comparison. The formulae

(2.9.) coincide with the corresponding formulae in

CI]

, apart from the surge mode coefficients which are not included in

Ci3

, if the integrals involving

_and.LkO

are

neglected in (2.9.) and the expressions of o(oj) in the

formulae in

Ci]

are deleted (these are the terms proportional

to for all coefficients and those proportional to ft for coefficients)..

It should be noticed that, as follows from.the equations (.32.F.), (33.F.) and (I. 4.31.), the coefficients and vanish on the parallel midbody on which NI= 0 . Besides for a ship

with fore and aft symmetry. the coefficients OJk, and a",L.) b"10become odd'functions of the coordinate and the

results presented in [2]

and 13]

imply that:

-6" '=O

x1 (

_0L(O4

= I(oi. '1 L.CoL

whence, it might be supposed that:

cA_I

-

"

-

=0

køt.,i

l.0L.t'i ko

for

kt21

L(Z

d t4

, with

the telations (2.12.) holding independently from the fore.and aft symmetry of the ship. However, this last result is not

confirmed by the expressions presented in

r3 ]

for

L

and tLi

, which correspond to:

a0,O

I _J. Zo1,1

tk I'

-2oz1. I I

- -

a.. _Q32..

for a ship with no fore and aft symmetry. Similarly, the terms:

=

0

S

(a!041-

_O4')

o,

vanish for the symmetric ship.

Hence for the ship with fore and aft symmetry the only non-zero terms in (2.9.) which involve the coefficients (2.10.) are:

(v

2

IL

x'

2o7..,Z (a.)

A53

-(2.4z)

A'(a'01-

_-Lo4O

Is

=

j

0

& ₌

a

oA,z.

-ji.x'E

_{(u-'F -('-l.x'k)i.}

_'41,2

-'2.Eg:LX'-

=

i.x' b3 Lx' &(K'.-

A

x'a33td

£x' -XF--

(

LX'/fl

i:i

L

If Timman-Newman symmetry conditions,

[4J

are to be ful-filled by the coefficients in (2.9).. it follows that:

-1

'k

'jk p

which within the scheme of the present derivation implies, as can be seen form (2.14.) m(2.16.) that:

L(Q2_

irrespective of the ship fore;:and aft symmetry, and:

-

= 0

(z.4)

at the extreme cross-sections., for ,

kai

3 , and

k 2. OLMt

L.4.

The relations (2.18.) are different from the results

presented in E2J

and

L3]

where nonz.ero contributionscorres-ponding.to the coefficients in (2.15.) have been found which fulfil

The above considerations show thatafurther investigation into the terms in (2.9.) which involve the coefficients (2.10.) should be undertaken. However, taking into account the results of

compu-tations presented in

[3] and Cs]

it seems that these terms

can be neglected in computations of ship motions for usual ship

forms.

According to the approach sketched in the Appendix F the oscillating parts of the nonlinear forces in (2.7.)

take the following form when expressed in the frequency

domain:

A

-

c

C-k

3Ttc

for k = 1,2. . . . , 6 and = 1 ..., 4, where

fkj

are

de-fined by the relations (11.F.) 4 (22.F.) and comply with

the heuristic rules described in the Appendix B. It follows that the components

_{Rk are}

represented in the frequency domain

(excluding mean forces) by the formulae:

A A

-(-Ak, +L13k16 +

_i,=4 1% 1-

2

/

.ç1 -i.. 0 '(U.

44t

.

foi-From the formulae (11.F.) 4 (22.F.) it is found that:

A A 'S ,' 2 LX'

(

p.. A

-

(I fa.}'

- (

r'

( I F

yt

.

ta-

+ p.. A 0

-

F.:%* -

(\c3A*

-(vL.5x' f''

1-- 1. -1

'. +

Ill

J 11z3'

p.. _{1% 'S.}

:7.L

(1iV..

(2.10) (2.21) A -.

cr+

A A _A

(fY -(kf

k _I A A A

-+

I

Vt

fj

\1- r ,

\t

:3Q

2.2) -( t

r,

+L*

- . t-r32..4, 1

/i=

(*

(4c4

4?:t

_52.

:,,_(!)(I+

-

lL

. 33

(r' Y

T3

A

From the formulae (2.20.) it is seen that the nonlinear

forces supply couplings between symmetric ship modes

3 and 5 and nonsyinxnetric modes 2,4 and 6. Another'

important effect they introduce' is so called frequency mixing by which a motion of a specif led frequency in one

mode induces forces of another frequendy (due to the nonlinearity) in the same and/or other modes (due to couplingsnot only nonlinear).

Besides, it should be noticed that the terms with the subscript T in the formulae (2.21.) correspond to the flow phenomena at ship's waterline and in particular they depend upon the rate of change of waterplane geometric

characteristics with draught. Thus new geometric characteris-tics of the ship appear in comparison with ship descriptions

based upon strip theory,E6 . . It is proposed to give the

*

name of Tyc effects to the terms with the

T suçript;

As a tribute to the late Cmdr. Antoni Tyc, Polish Navy officer in the West during the World War II, and

ship designer,manY of whose ideas about the influence

of ship. form upon seakeeping qualities have been

confirmed by recent investigations.

A A

T5

₁

-

-

f

If:.+

X

3. The Equations of Motion.

In terms of generalized force components the equations of motion for the ship oscillating as a rigid body under the influence of hydrodynamic forces, take the form:

kMkHk°

(Aj

with the values of the index. k = 1,2,...,6 denoting the force modes corresponding to the ship rigid body displacements.

In the equations (3.1.) the symbols

F

and represent the generalized components of the inertia and weight forces respectively, which can be expressed in terms of ship displace-ments, velocities and accelerations,.to any desired degree of approximation from the formulae (i.C.). and (2.C.) of the

Appendix C.

The symbol signifies the components of the resultant hydrodynami.c force acting on the ship. Within the perturbation

scheme employed in this report, these components are expressed

as:

for k='112-,...1.

The indeces of primed summation in the equations 3..2) assume the values 0,1 ...,4,7)which denote the cross-sectional modes of motion. In comparison with the equation. (2.3.) the excitation

mode 7 haS been introduced he.re. Once. again it should be

pointed out that the cross-sectional modes 2 and 3 split into ship rigid body displacement modes 2 and 6, and 3 and 5 respectively.

For the consideration of the equilibrium of forces (3.1.) it is useful to separate a.verage or mean forces from purely oscilla-tory forces. Hence, the component

_k

can be written as:

w_=

where the line indicates mean value and .(by analog.y to the notation in (2.19.)'), the superscript denotes oscillating components of zero mean value.

The definition of mean values is natural for periodic

or limited in time phenomena but implies some limit process in other instances.

In the perturbation', scheme the decomposition of the hydro-dynamic resultatns takes the form:

and: where: -P -t 5_ . KL\3

k=

1z

-F... -P 1L.1

(kjL

0 Lcoo

*

L'

_(kIj

- <oL' t.=1 -F

+

The terms in (3.6.) that involve the index 7 of the excitation mode represent components of the hydrodynamic exciting

force, with the forces due. to the interaction with ship

motion's included,. By comparison., it follows from the equations (3.6.) and (2.6.) that:

_ with:

and :-.

Fk

4t1k

In the above equations it has been consistently assumed that ship's displacements, veloctities and accelerations do not contain mean components, this corresponds to the definition Of the reference configuration in the paragraph 1.

It follows from (3.1.) that the equations of motion can be rewritten as:

t _I_t+(

-'Hk k _{'H. - '} I (3.9) (3.40) for k = 1,2, ,6. Hence, the equation (3.10.) represents the equation of oscillations with respect to the reference configuration. By taking into account (3.7.) and (3.8.) it becomes:

r

_F4

1

-

-for k=1,2 ...6; or after separating the linear part (compare with(3.6.)

-t-.. ) =

--

-

(iL.

-for

k'l,21..1&.

For steady oscillations the equation (3.12.) can be put in the frequency domain In the form:

k

-A

for k=1,2 ...6, with P] denoting the representation of ship rigid body displacements in the frequency domain.

(ki M

Ct1

A'c(3t1,

with all other terms of the matrix

_C9w.

equal zero. Inthe equations (3.17.) and (3.18.) M denotes the mass of the ship. Besides:

+M

(3.4 )

tht follows from the equations (2.20.) and (44.F.), (45.F.), that: ...

(q(3)

fov

k'121

where:

M-i2-

(pt.. Prjt

+ Cki

4

_Ck:

k)L.=12.1... c:',

and , and C k are determined by the

equations (2.9.) and it is found from equations (42.F.), (43.F.) and (3.14.) that:

I

'

0 0 0 _o

_-,..

_o o

₀

0,

z'c.c

0,

L1)x

lcJ-

0

-X7

o X'CG.

0

0

0

0

0

48)

g-

(.5I2.A

-'k

= -ç L Z!6 f

1.

I-

L

M

' "P1

The components in (3.15.) are determined by the

equations (2.21.) from the equations (16.C.), (]i7.C.), (18.C.)

and (19.C.) it is found that, in (3.15.):

3.2.0)

for

k

4 and L 4 2. and

I'

A

FM3

@.24)

F.41

4r

-

-14r

(1

.wit'h all other nonlinear mass forces in. (31.5.) equal to zero. If the assumption is made that a small parameter . corres-ponds to the excitation mode 7, it follows that for stable

ships:

O).

zz)

Besides, generally, the mode 7 should be condiered.as com-prising even and odd component's in the scheme of Appendix B., Hence, according to the scheme it is found from the. equation

(3.8.) that:

IC:?.t

O(O%P}

=

-t

in the equations (3.15.).. The equations (3.15.) are given in the explicit form in Appendix G.

a) the equations of motion are inherently coupled and

non-linear,

b') the lowest order terms in the equations occur for the

modes 2, 3, 5, 6 and as the inertia term for the mode

1, and are' of O(& . They do not.involve any forward

speed effects or nonlinearities.

in the surge equation (mode 1-) the lowest order term is the inertia term of 0 (9,) whereas the remaining

terms are of 0

the roll equation '(mode 4) consists of terms of 0(E)

and does not, involve explicitly forward speed or

non-linear effects. However' it comprises non-linear couplings with the modes 2 and 6.

the linear terms of 0(a')in the equations of: sway,

heave, pitch and yaw (modes 2,3,5 and 6 respectively) are

of 0(&Sh

(forward speed effects) and of 0E.)

(couplings of heave and pitch with surge).

the nonlinear terms occur in sway, heave, pitch and yaw equations and are of O(E.ij-. They represent hydrodynamic couplings of the modes 2, 3, 5 and 6 between themselves and with the mode 4 as well as inertia couplings of'

he'ave and pitch with roll.A'r'art of the coupling terms

re-sulfrom the interaction between radiation and exciting

flow phenomena.

The form of the equation (1.G.) suggests the solution for surge in the form:

. ,..

which if inserted in '(l.G.) yields:

0 (.3.35)

and it follows that the coupling terms of heave and pitch with surge in the equations (3.G..) and (5.G.) become of

0 (Ei)

and drop out as 'being of o () . The main defi ciency of the. equation (4.G.) is usually considered to lie in considerably inadequate prediction of roll damping, C7J

this results mainly from the factthat roll damping is

main-ly of viscous origin whereas viscous effects are absent

-Taking into account the influence, of bilge keels the poten-tial flow damping is also not easy to estimate occurately.

The usualwayto correct roll equations of the kind (4.G.)

is to determine the damping term (roll to roll damping) . by an empirical or semi-empirical method. The damping is found

to depend nonlinearly on roll motion, which difficulty is. circumvented by employing quasilinear roll damping coeff i-cient which is considered to depend on roll

ampli-tude. As a.rule the coupling terms with sway and yaw are ne-glected. A comprehensive review.of such methods is presented

in ]. If one of them is applied the equation (4.G.) can be rewritten in the form:

(a3

*

2J?,'Pt

_{2'c&hVj. E}

+

-

(A

[(M

J

4'642f

Although a separate analysis is required in order to determine the order of smallness of the quasiline'ar damping terms within the. scaling scheme employed in the pres.ent report, if the

simple formula.':

is used for a hydrodynamic force, with

5L

and.

it follows that the terms involving essential'non-linearities in (3.36.) due to component may be of

O(.

This shows that potential terms of can be significant for the accuracy of the equation (3.36.). Although these terms have not been derived here it follows from the

previoüs.. considerations that;

which is confirmed by"the results presented in C 3

Besides, of the lowest order nonlinear hydrodynarnic terms, their remain , if the exciting terms are

left out of consideration. The formal expressions for these,

and,,ma'ss and weight nonlinear forces of in the

equation (3.36..) can.be derived according to the approach presented in the Appendix B and C. An important.factor intro-duced by the nonlinear forces (not expressed in a

quasi-linear fashion) is that they produce frequency mixing pheno-mena (see comment in the paragraph 2). The failure to do so may be one of the major disadvantages of the quasilinear

approach.

In this context it seems worthwhile to point out the possible influence of nonlinearities in the sway and yaw equations upon the roll motion. The equations (2.G.), (3.G.), (5.G.) and

(6.G.) impI - the solutions of the form:

ti

t

(3.3

for i = 2,3,5,6.. Itappears that the nonlinearities in the sway and yaw equatIon induce the higher order terms and

injV

which In the linear coupling terms of the roll

equation produce couplings of providing another mecha-nism for frequency mixing. Due eQ the same couplings the

terms Sin and Sli introduce forward speed effects of

Finally the comment in the point a) above, about the inherent nonlinearity of the equations (.1.G.) (6.G.), should be

elucidated. Taking advantage of the flexibility of the order of smallness relations scheme introduced in Part I,

it Is possible to assume the following relation:

= o(.Sk')

(3.'o)

and find the nonlinearities in the equations (2.G.). (3.G.) and (5.G.), (6.G.) to be negligible in comparison with other

terms. In words the relation (3.40.) can be described asa high

frequency - small motion assumption (compare with (1.3.33)). On the other .hand if rough ranges of the slenderness para-meter and motion amplitudes are considered e.g.:

and it is agreed that in applications for

3i <<1

it is

enough that:

S <0.5

the relations (3.41.) produce:

e'o, .62.7,

(.43')

according to (1.3.28)., and it follows that (3.40.) is not generally valid.

In connection with (3.41.) it is interesting to observe that in terms of roll amplitudes it becomes:

vt4e<

O03O0?

as can be found from (I. 3.29.).

The relations (3.41.) and. (3.44.) indicate how the amplitudes of pitch and yaw compare with the amplitudes of roll on the basis of equal displacements of ship's hull surface, and how such a comparison is influenced by the slenderness

4. Conclusions.

The elements of the equations of motion de.rived in this report indicate that, following the perturbation scheme of Part I, the third order equations are inherently nonlinear and.can be

reductedto a linear se of equations only if the high fre-quency - small motion assumtpion is adopted (see relation

(3.40.))

The frequency domain version. of the linearized equations coin

cides with the corresponding expressions presented in ti]

if the terms. involving, forward speed dependent cross-sectional radiation force coefficients are neglected .in it (see equations

(2.9.) en (2.10.)). as well as the higher order terms in the expressions in (ij(.i.e. the terms which are proportional toU2,

and to U in B coefficients).

4i.

Although neglecting the speed dependent cross-sectional.co-efficients seems to be justifiable when ship oscillations are concerned this may be not true in the

instance.

of the evaluation of cross-sectional forces,. as the results presented in ti5]

strongly indicate. The discussion in paragraph 2 suggests that further investigation of these forces is necessary.

The nonlinear terms in the equations of motion provide couplings between sway, heave, pitch and yaw modes, and between these

modes and the roll mode, although.the roll equation.remains linear (see Appendix G). Apart from invalidating the usual

separation between lateral and longitudinal motions, occurtiiig' in the linearized equations, the non-linear terms can produce so called frequency mixing phenomena in. the frequency domain,

whereby the motion ofa specified frequency in one mode

in-duces forces of another frequency... in the same and/or other modes (due to couplings, also linear).

Besides a part of the nonlinear terms results form the wetted

surface varying with time, which In comparison with the linearized

equations, see. (6.], introduces the ra.tè .of change of the waterplane characteristics with respect to draught as

corn-plernentary ship form parameters relevant to ship's behaviour. in waves.

The third order roll equation:Is linear but it involves coupling

terms with sway and yaw. Although the proper assessment of additional terms due to viscous effects should be. carr.ied out by including the viscous effects in the perturbation scheme,

the simple approach illustrated by equation (3.37.) indicates that the resulting terms should be of fourth order of

small-ness. The importance of viscous damping terms is very well

known and it follows that the fourth order potential terms may also be of interest. These can be derived according to the scheme presented in the Appendix B, however the inclusion

of viscous effects in the perturbation scheme prior to such a step would be advantageous. The quasi-linear approach is useful from the practical point of view, however one of its drawbacks is that it excludes possible frequency mixing

Ref erences.

ri

Nils Salvësen, E.O:. Tuck, Odd Faltinsen,

"Ship Motions and Sea Loads", Trans. SNAME, vol. 78,

1970.

E21 T. Francis Ogilvie, Erest 0. Tuck,

"A Rational.StripTheory of Ship Motions:

Part I", Rep. No. 013, 1968, Dep. Nay. Archit. Mar.Eng. University of Michigan, Ann Arbor.

C3] Armin Walter Troesh,

"Sway, Roll and Yaw Motion Coefficients Based on Forward-Speed Slender Body Theory" - Part I and II, Journal of Ship: Research, Vol. 25, No. 1, March 1982., pp. 8 - 20.

L4 R. Timmand and J.N. Newman,,

"The Coupled Damping Coefficients of a Symmetric Ship", Journal of Ship Research, March 1962, pp. 1 - 7.

[5:1 Odd M. Faltinsen,

'!A Numerical Investigation of the Ogilvie - Tuck Formulas for Added-Mass and Damping Coefficients", Journal of Ship Research, Vo. 18, No. 2, June 1974, pp. 73 - 84.

[63

J.S. Pawlowski,

"On the Application of Nonstructural Models to Ship Design", International Shipbuilding Progress, Vol. 23, May i9'82, No. 333, pp. 125 - 135.

[7] Yoji Himeno,

"Prediction of Ship Roll Damping - State of the Art", The University of Michigan, College of Engineering,

Report No. 239, September1981.

18) T. Francis Ogilvie,

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

i:

i

Lothar Collatz, Julius Albrecht,

"Aufgaben aus der Angewandten Mathematik I", Akademic Verlag, Berlin 1972.

[ioJ

Conrelius Lancros,

"Applied Analysis", Prentice Hall, Inc. Englewood Cliffs, 1956.

[ii] J.A. Pinkster,

"Low Frequency Second Order Wave Exciting Forces on Floating Structures",

H. Veenman en Zonen B.V. - Wageningen, 1980.

[12] M.A. Abkowitz, L.A. Vassilopoulos, F.H. Sellars,

"Recent Developments in Seakeeping Research and its Application to Design", Trans. SNAME, vOl. 74, 1966, pp. 134 - 259.

W.H. Livingston, D.L. Newman,

"Advances in Implementing Ship Motion Predictors", AIAA 18th Aerospace Sciences Meeting, January 14 - 16,

1980/Pasadena, California.

[14] Louis A. Pipes,

"Operational Methods in Nonlinear Mechanics",

Dover, 19.65.

[151 G. Moeyes,

"Measurement of Exciting orces in Short Waves", Technische Hogeschool Delft, Laboratorium voor Scheepshydromechanica, Report': no. 437, June,1976.

Appendix A.

The Estimation of Integrals over the Image of the Wetted Surface.

In order to be able to evaluate the forces and moments ,it is necessary to express in a suitable form the

integrals on the right-hand sides of the formulae (1.18.) and (1.19.). Taking into account the cross-sectional character of

the veloctiy potentials determined bythe eqautions (I-4.3O)

and (I. 4.31.) and that' beam models are usually employed in hull strength calculations, a parametrization of the inte-grals is conveniently arranged with the differentials directed along the contour and along t:he axis respectively.

The assumption is made that almost everywhere on the wetted

surface the points belonging to it fulfil": one::or both of

locally valid'' equationsof the type

S,( i..j 1z.)

- S ,,

(uc -z)

0

The differentials of the surface area can then be expressed in a vectorial form as:

d5

d A

with d representing the differntial directed along the contour in such a way that.the fluid domain remains on the right-hand side, see fig. :l.A.

Figi A

*)

itis supposed that a finite number of maps of the kind

The differential d.c can be written as:'

ct

dc

with:

and:

dC. [o1NJ,r'.j3')

(0 d-!1 ciLj').

_(NA)

The differential

dt

can be represented by one of the formulae (6.A) according tothe validity of the representation (1.A):

ciC{

d(i1%fo

see Fig. 2A:

yL

A

x

Fig. 2A

From the formulae (2.A) and (3.A) (6..A) it follows that:

c-NJ '!(\JNJ3')

1

SL

_I

where one of the vectors in the brackets should be chosen as

in (1-.rA) and (6.A). Besides, from the formulae (1.4.4) and (LA). it can be found that:

-

I

A) 4 L

dsL

-

)c

C,

with D and

b

defined respectively by

bL

±

.

(5LZ..

Hence

d3

can be represented in the form

-

I

(W4N,tsi1\J'3

d

chxd

L

(rj1 F'JLI tJLI

L (-\I11L

di'1 dt' -

ct)

The introduction of the normalized surface differential

_d

such that:

froSi\t

d1Ld

leads to the expansion:

with:

=

(io.A'

the positive sign applies. when the "positive" side of the surface pouiarLg towards increasing independent variable

(3or

. respectively)

d

(oI

_{(o, rJ1}

N1 de

(O

I

0 O

- ci

o 0)

=

(m.,d''1

_0%O)

C

c&4'

o

with:

The above formulae do not cover the case of

I1 I

'1 ,

on a part of the wetted surface which is of nonzero area. In order to be able to include itegrtions over e.g. transom stern, the dii.. differential on such surfaces is introduced in the form:

dt

n1 (0! dj

and from (2.A) and (3.A)it follows that:

(NIoo

(5.k)

-in the normalised form:

dc(d1o1o).

Hence,, if the presence of a transom stern is taken into consideration, the formulae (1.18) and (1.19.) can be re-written as:

+

.c._

-1b'

d

-with and denoting respectively the coordinates of the

bow and stern cross-sections, and

and haignifying

contri-butions at the transom stern. Besides,

() and(); denote

so called hydrodynamic cross-sectional force and moment corres-pondingly. They are defined by the following equations:

with:

d

fE

((DLI0tO

J

t

f(twdcO.o)

(tLd.A11 OO)

The transom stern contributions are as follows:

S

].ddT

1' +

_d

) with: t A "A J C' + £- _L

' 6

c

at a')

4;

-$ E(+

crk

*]d'i.

-J ('&dE,

L=-+prr-'A3dC-d

It should be pointed out that the expréssions(i7.ftj.) and

(18..) are of rather formal character since the potentials defined by the equations (1.4.30.) and(I. 4.31.) have been

formall equated to zero for the cross-sectional contour

reduced to a point, as it is at the bow and stern cross-'

sections. For an adequate treatment three-dimensional flow

pattern should be estimated at the transom-stern, perhaps according to the approach outlined in

_r°J

for

the flow at the bow. This problem, however, will not be

further considered in the present report.

For the evaluation of the expressions (18.A.), (20.A.),

(22.A.) and (23.A.) a typical cross-section is considered in its reference óonfiguration.. As shown in the Fig. 3.A1

the oscillatOry motion of the ship carries a point on the cross-sectional contour to instantaneous locationin space.

n

Fig.3A

It 1s assumed that in the. displaced configuration the entire

length between two pointsQ and Qon the contour is in

con-tact with water. Due to the smallness of the ship displace-ment and wave elevation, each of these points remains close

to the corresponding point on the contotir at which the

cross-section meets the undisturbed water surface in thereference configuration. In order to determine the image of the wetted surface on the reference configuration surface, it is enough to find the images QL and of the points and Q.

respec-tively, on the reference configuration of the contour; for

It is assumed that at the points

and

:

O(1'

which means that the slopes of t'he ship;sides with respect

to the vertical are bounded at the water:iine.

.

The points

and

on the reference contour can be defined as:

6Z

ç['(.') +

(/Th= 0.

Thfollows from the Taylor's theorem that:

?2

;' .

'i

-(o.

=

(

_'r'

cr')

(26.A

where, as can be confirmed, by consulting Fig. 3 the values of are determined by the solutions to the equation:

z

-

'+)]j

-&

)

with 1 1 I , and under the assumption that the

denominator on the right hand side of (28.A..) is different from zero, which is justified by the order of smallness con-siderations.

Following the Newton's;

method, see e.g. tJ ,the first

and second approximations to are respectively:

IL

i[t-

cz/

_:J

and it follows that:

- O(oP'')

Taking into account (3].A.) and (34.A.), it is found that:

c'

r-

_eLc'+)iI

-t

_L35.

with,:

t

,,: ts

tri: -=

+

&=

The denominator In (30..A.) can be fuither expanded, producing:

=

[('+-

_1Lx

-('.6c)] I

+ [

Z' -()1

-I.

O( 1.1 IZ)J

The second. term on the right-hand side of the equation

(29.A.)

can be expanded

in the form:

t,Et1i -(tJij

= k.

41 + OtQc61)1,

with:

Hence it follows that an asymptotic expansion of

can

be found in the form:

-4-. =

or:

'+E(t6-fll-to be approximated by:

*1;

(3s))

_z'[f.(Jit14,ty+

" £

(i1D

[fCil1W)

A] I-F o( 1 &-'+ It),

where:

The:re are two special cases of the integrals of the form:

04..

(,)od

SL-.-flo(/Ad

3o.A)

which deserve further consideration in connection with the. derivation of appropriate- -formulae for the forces and moments. According to the relations (12.A.) and

..(13.A.).

the differen-tiaidC in (41.A.) and (42.A.) can be expressed' in the form:

)The circle may denote scalar, vector or ternpr product,

applicabl.e according to the tensor rank of

f

The expression (36.A..) allows the integrals of the form:

(tIItW

Ll.iftS

11L2,I'2)dIj/

Hence the integrals (41.A.) and (42.AJ can be rewritten

respectively as: and: -

-

_£

d'_d

SL

- a' i'

= and:

it Is found from (44.B..i and (45.B.) that:

-t

t-/s

1Li

-

(4sA

(.

A

A}

.A)

L

Ad

+ ofl.

ç

I

-it ) cLi'

)

The above derivation of the formulae (48.A.) and (49.A.) constitutes a more straiqhtforward alternative Of the derivation presented in the Appendix I of

t-J

The formulae. (49.A.) can also be rewritten in the form:

-o((iIo1o)Ad

°=

(5o.

Appendix B.

Asymptotic Expansions' of the Cross-sectional Hydrodynamic Force and Moment.

In order to facilitate the presentation of the quantities

c),

_C)'

DA and t1p , which are defined by the

equations (19.A.) (24.A.), it is convenient to express them in the respective forms:

with:: and: with: with: L*

t8f

E-

$T)Gc.)

e

'dd

)

1E_4

( *[P'AR

(i3)

cc

&) -t

d

I

'+)à

].dd' (.a)

-s

=

&e çc'.

with:

A( t4

tj. (')

ii

S. i- ('*

&) ('

.

.-.') AR

c

In the equations (6.B.).and (8.B.) is determined by the

formulae (2,4.A.).

By taking into account the formulae (37.A.), (38.A.) and (39.A.) it is possible to rewrite the formulae. (2.B.) and (4.B.) as follows:

=-j

_

4t1'1-'')] (

-t. Of

(L

..

(

) ].H

+

o(1y\t2

+

*

(t'.I1b

-

d/

[

LN -4.

t,

t)} ( . 0_ + _o

_I

and:

with:

D?)

_(ii.t

for and respectively,

a{'*[

+ T)

* '/AJ 3

E

for Aand

respectively.

According tothe formulae (22.Aj, (23.A.):

J C()

7J

d4'

?±

'

for Iand lf.,afld

-./

_-1

±=

(

* £1.')

J,

'A?j

dAç'

-1 ₍

_')

_{A R}

'

_,

for AA,

correspondingly.

By means of the formulae (9.B.) (14.B.) the asymptotic expansions of the cross-sectional forces and moments can be found from the equations (1.B.) 4 (8.B.) if the approptiate expansions of the integrands are substituted.

=

-The expansions of R and v were presented in Part I. From

with: =

j

;zrI

*

Lf-j

+

* L3

(..,--r

The corresponding asymptotic expansion takes the form:

f'-t

-+ with: -₍₄

-= -

-

2'+

_i")

C) -

(a--

L-

'+1

Besides from the equations (1.4.24.) and (1.4.25.), it follows

that: L4\

o (

o_)

Is

and. according to (34..): (-:: '(1 _/?: z 0.

where:

and:

fTE&7,3FA

$

x(/t1o1O) (&L) [cbwd'1o10

0 J

where:

=

3 -

rt'

(zi.a

as follows from (1.1.4) and (1.2.13.).

From the equations (2.B.), (9.B.), (1O.B. (21.B.) it can be found that:

(-

(i)')

+.

1f

. C C cL3 )

/

with:

-

-'.+

'Z'

ç:! dr', -d1-") (1i.B. ),

(17.B.)

+

(a)+

1--cLif) *

: 4 x 0

-c')

4TV:C13+

- '

OvL1O +

'

-d..9' +

+

V[ (.,

(od- d')

t (. + 1 ' ')

(\d

QtA', -d")

± (i-t'2

x ₍₀ Jc _d41")

*

/

L'j'-"\

I..Q ' '3

1)

with:

andM

denoting the cross-sectional area at

(-@

with signifying the first moment of the cross-section area with respect to the z'axis at Besides in deriving

(25.B.) the relation (48.A.) has been employed.

From the equations (2.B.), (1O.BJ, (11.B.), (17.BJ(21.B.) it

is found that: fr() ₀ (g.9.T) =

{#-E ((ç

vi2t_

-r O.133) l

-T(--

_{oIto -j}

)C

1O%O4j 4

fl

A'

i-1e

(j4

7s'

*

-

(L1)f[C.t

-I-

"-j 3...

-

lx. * _.j-

(o,o 't.)

Next, the equations (2.B.), (8.B.), (9.B.), (12.B.),.(16.B.) (2O.B.) lead to:

'- _'-05.)

AfA. =

lfi

_FF

= 0

C31.

and:

-('t3--'ts'

*

where the subscript A denotes corresponding quantities

esti-mated at

From the equations (4.B.), (9.B.) (iO.B.) and (12.B.) it fol-lows that:

and:

Pr =

(O&)

A Aç

-t T 7'A

('L1OO) -

('t)

(iLOO)

-,

with:

(35.1)

where rx')denotes the first moment of the cross-section area with respect tothe axis.

Besides:

Wt.

Y'i '

1?/rQ

I

,'.- (3)

and:

- ₌

0-Tãkingi into account the formulae (35.D.) and, assuming that the reference configuration of the hull is symmetric with

respect to-Lthe prane

0

, it is found tha1:

-

*

(

Sc-+3o.o,dA.1)J.

Th')

-Besides, for / expressed in the form:

)

1-oQ';

3 '

C-Dd'

oO)J

5 £ (

( o,

th', o)

- 44. ),

(o4d'O)

-,,

)

)I (o1,L1o)+(

3(o1d o)+1

(/j_t.

-p

_3;tt

_',

(01

c%'

O)4

S

(0

*

IkLIz.t j4-+

(04 dt'o)

+ _/1aL.

(J(o1o,-d.4') * st.1[ ()f;

(o.o-a')

-')(+ '4t')II

+

+p0)3(o1o1-cL')

+ of +

()[J( (o1o4d') +

j'p

-(°(* (o10-d.q') +

+ (3_

5X(

+

i

+1ejL ₂ aj

(°

0-

d.

*

(o.o-d')

--'4

(o,o-d').

Similarly it is found that:

tz(3-

5')(

4)*

(3.8

- _vl4

-

(v3

-

₃ 0)

(ii'

)

+i[

( (M' J..

At

+

z(

-'k5 ')

(3)+

T6AR,o.o)_

, o

o)

-

E.. (( 4'd'

₄

4

in order to sum up the above results it is suitable to adopt the scheme based upon the distinguishing

of five

modes of

the motion of the coss-section

Ot1...j

hi _{, which} corres-pond to the steady forward motion, surge, sway, heave and

roll (of the cross-section) respectively. The fluid

flow

due to these modes induces gener.lized cross-sectional radiation forces for

k1R1..1

,which.represent components

of

the

cross-sectional. force and moment with. respect to V on. the

cor-responding axesof

_the.(19,1Z.

system

of

reference. It is possibi.e to express in the form:

;;1b')k1- (r

_k'

"-(3)

Af. =tT(3-AA(4LoIO',

where denote the delta type contributions fromthe.

end cross-section, which after integration with respect to

d'

give the and t1 terms in (18.A.),compare with

(5.B.) and (7..B.).

According. tothe above. deivations:

4 (.

°3tv1(

_Dd1-7) ...

(-"A

denctes the nondimensional Dirac's delta

"function"

for the integration along the ship, the. cor-responding

(f

X'A has the dimension of

!

(j'1

.çz (,d'

¶

4a

Mc; °('d'

L

54=_)('f34

-.

',f

f_. F. . F.! rug P ,H

-

Sh-l-

*

.I_

'+J-Li 4.Ot. .zot .p

with;

4E

r4

.2

T4I:O2

_°%d'

Tzo

fZ.o4.

S'h

+

t

with,:

'vQ=

.i _r0

_'

L

,

(cr)

4- -r_.-I l

+

'c-1

I,-'.

/5'_

4)7_

=

t

7:4

i-ri

2'

1t

4ict

Vt:

cz:

-c'

t;:Q/ç

r

(2Q \

-

-

'

4&

':

Y = iL

:i).0c

:tj TM

t::L

t-r--'I -3 ze,0

:t(4

(Q

(,'L

+t2A)

r_rt' 1) Ot:J

-TTM

t0ç3 4c:ot

Y7ori:jx

7O

o.,X

0 (

toc1

with:

('?!

c1'

with:

cZ3

+

3

%d1

2 [ (t)

4-T1] t

p%-+3j+ vi6t-/)c 22.1

2.Z.*

1.

i3

I,

_*

(ss B)

8)

2

cç

;-

i-4,')

2jt]L(

P_gig

-

_-

p.--':

i

((\

\t.

t13

r

with:

33=

33

(,3r-t (r-i (-di')

(,)ç±zsJ..-a')

(Lg\1!L ,:1ti

fT33

_i

_{L zr- 3tJ43}

24

h1')

+

+

,- //

i-i::

I 324 t3211

t3z4

with:

7/

(9_\

O+( 132A

bç0

d'),

cL

O+

-

',')

,-cL'),

:

L*

'hi,

L7ç,

;42.44:)L a3"L

=

--

1

(4B)

with:

44

-Ia O 2.

f34

r

(- d')

T3,4

i

, +

p2:t}

I

f

52.Z

-533

34

-

'-/

_{_34}

h23

\)ç2L

(L s)

with all other components in the formulae (46.B..) vanishing

up to the terms of 0 ()

It is useful to notice that the expressions (4'6.B.) 4 (67.B.) can be described by the following heuristic scheme:

a) the force modes correspond to orders of smallness according to the relations:

i'-0(1'),

b) the motion modes correspond to orders of smallness according to the relations: o '1

'-' O(j) /

(e5.1)

-

₎

c) the order of. .smaiinessof a term in. (46.3.) is equal. to.

the order of smallness of the produc. of. the indices times e.g.. :

O(v

odd terms are identically equal, 0., e.g. 1 2.33

_

0

c3

9 2..

.0 (:tt

=

and

the force modes correspond to pair.ities according, to the relations:

O'i35

"-'

odd

the motion modes correspond to pairities according to the relations:

O.L4,3

"-

f.Aj,YL.

,#

ocU

f). the pairity of a term in (46.B.) is equal to the pairity of the product of the pairities of its indices., according

to the rule.

odd. x even = even x odd = odd

(34

Appendix C.

The Weight and Inertia Forces.

The weight and inertia generalized forces acting upon the ship are defined by the following ecivations:

M

for the weight force and the weight moment with

respect to P with denoting the acceleration of gravity and _H signifying the mass of the ship assumed:constant, and:

iE=

4dfr1

for the inertia force

and moment of inertia M,with

res-pect to P

, where is given by the equation (i.1.4.)

According to (1.3.) the normalized forms of the equations (1.C.) and (2.C.) can be rewritten in the form:

-'1

(Q1O1'fl

A

where the following relations have been employed:

N

L1dcB,

dM

c)Md

dQ

and:

dç (')

MP

Only time dependent, "dynamid", parts , of

and

_M9?

respectively,areof further interest. Besides,

bearing in mind possible applications for the estimation of structural loads the forces should be expressed in cross-sectional form.

By taking into account the relation:

=

(.

%''

-with

A

defined by (27.B.), the following asymptotic ex-pansions of the cross-sectional generalized weight forces can be found:

19t

.-.

'c3?D +

(.3))

where:

with the following definitions adopted:

Mc%AM(

(9.c'

where: and : with: -. (2.

-('

1--+

o(Q5')

(itc)

A '....,

'fl"

ZGYL

I

-'L

), pp_1 C3)

'$

\-, p. :1. M + AM

-'-

'-

(3 = 'v-n..

+

(- )

(A3.c LtC)

*.

-I-jt.(

-x'ir'G.+

(-J('o1o) +[4

1(,+J(o

-DjíA1,

(42.c.')

and denoting the mass per unit length at the

cross-section at

, denoting the mass per

unit length moment with respect to the Z and axis respec-tively.

Similarly the cross-sectional generalized inertia forces can be expressed as:

and AM(x')denotes the cross-sectional mass per unit length moment of inertia with respect to the c' axis.

it should. be pointed out that the above formulae are. based upon the assumption of , z',. , and -M(I')all being

of _O(1 . In particular cases of mass distributions, deviations from this assumption should be considered by taking into account a technical inte'rpretation of.the order relations.

it is convenient to sum up the formulae (7.C.) 5 ,(9.C.) and (11.C.) S (14.c.) in the forms analogous to that used for the

formula(.46.B..).

Hence:

cg

fb91'tk

k=

4z1

...

&

.

1

and. it is found from (7C.) 4 (8.C.) that: where:

with all other components in (16.C.) equal zero up to the

terms of Similarly:

1k'121.1

/7'

* I

M

£

-

'

M I'-.

'

_-I

f

-Ll'i

Ic

-

,4.*

_\

4

1'2

t1A

-It 5 4

ct15.

-p1 -.-I

*.w \

fM,=-''

--c'

61k,i

-s

chS

_LLA)

''AM

-th

-

2(+ x/;:'GM

£Z(

_{-'k ''4}

.fr*

_I

%/

AM

£

*.*. p1 ._./ I

and all other components in (18.C.) are equal to zero up to

the terms of O(.i").

It is worth noticing that the rule c) of the Appendix B.(for the formulae (46.B.) 4 (67.B.)) applies to the order of small-ness of the components of generalized forces in (17.C.) and

(19.C.) if:

the force modes are considered to correspond to the orders of smallness according to the relatidns:

"-

0(A)

(zo.c)

Li

'-the motion modes are considered to correspond to '-the orders of smallness according to the relations:

The change of the orderof smallness of themodel ,both for

forces and motions, in comparison with the relations (68.B.) and. (69.B.), reflects the fact that the smallness of the slope

of the hull surface with respect to the

axis () does not

effect the weight and inertia forces, whereas it effects the hydrodynamic forces according to the adopted perturbation scheme. This shows one of the singularity properties of the

scheme.

It is also important to notice that:

194

(zz.c)

and:

13'4

O('vL)

(.c)

and all three expressions involve so that the nonlinear coupling terms and cannot be neglected up to

the

O(i)

on the assumption of '=o(1), apart froth the fact

(e.g. balast conditions). This is because that assumption

would lead to neglecting as well, which is unacceptable for the proper predicition of the roll motion.

Hence the.oniy wayto dispose of the nonlinear inertial

coup-ling between roil and longitudiiial motions is

to

assume that: (4.c)

The coupJ.ng forces involving are not so important since besides the possibility of assuming that:

'3Tcr

oCE),

(a.5.c)

they vanish for a ship with lateral symmetry when integrated along the ship.

Appendix D.

Memory Effects and Classification of Potentials.

In paragraph (1.5.) memory effects connected wth the boundary problems (1.4.30.) and (I. 4.31.) have been

ds-cussed. In order' to proceed with the estimation of radiation forces acting on' the hull surface it is necessary to classify the solutions of those problems in a suitable way, and to discuss the memory effects further.

The technical motive in the classification of the potentials is their.respective symmetry or asymmetry for hulls with late-ral symmetry. Another and deeper motive .i the distinction between the potentials according to the form of the corres-ponding boundary-.conditions.

A classification following this criterion has already been

introduced by the.definitions (I. 5.4.). That however does not go far enough since the separation of the time and space dependent functions in the exciting boundary terms, as in

'(1.5.1.) can not be carried out mall cases. For the sake

of completeness some of the derivations presented in the paragraph (I.5). are repeated here.

First to .be considered are the potentials determined by the boundary value problems of the type:

The solution to which is sought in the formi

compare with. ..(I.5. 1. ) and4L.5.6.j.,...where. the convolution

integral on the right-hand side of (2.b) represents so called in D, at x (,4.i)

on z = 0,. in D, at x

on

C0(x.

memory effects.

The boundary probj..erns for and are found (see paragraph (1.5.)) to be of the form: ( _;i'r( ( = 0

0, ctb

x b) (p (,S) (11 ,1. with: I

D1 for

for-

'o.

It has been shown in the paragraph.(i.5.) that the potential can be expressed as:

jE)

di..

where each of the is determined by the formulae (2. 4 (4.D, in which*):,

I c3?

I

tj't'J3-z'

Similarly it is possible to express the potential

has been slightly redifined here.

Lv1c) as: (&D) b)

(NiJ

(1

i..p())=. (q(q).10

c0,

'0

d)

'=O

o. :z

0

0,

with the initial conditIons:

The potential can be written in the form:

IL c

and it follows that.:

then in.(3.D the condition d) should be deleted and it is found that:

f

':?O,

1.D')

and:

c=

!4/ (.2.D')

as determined by the equations (3.D)a,b, (4.D)b'änd (1O.D).

For the potential , which is defined, in the

equations (1.5.4.), the exciting term can be rewritten as:

U

NL-

x"( N-

+ (43.bs)

+

o -

(_5x"j(

NL

oEJ3

'(Nt

Hence the potential can be expressed as:

(5)4

c)

where each of the potentials is deter-mined by t'he corresponding boundary problem of the type (1.D.)

-0(E .Y'5

s4(t

(+yx'

w-31

')

1A44

(E1A

with.:

rft10.

ks"'

It should be pointed out that

, other

ptentiais

exciting terms are separable,

31(Q') (N3a').o

(q[-3-

-i(NL-z. +

1

' (N+ NJ320

In a similar way the part of the

.zgi

potential,

corres-ponding to the exciting term:

w-(q ;)

NJ3],

in the equation (1.4.3 1)g, can be written in the form:

from the remaining part of. can be extracted for which as follows from the expression (2.D.) applied to the potentials in (5.D.).

This however cannot be done for the whole potential and therefore it is necessary to consider the boundary problem:

on z = 0, in D, at x

(N

N3

on

C0Cx,

in which

t4(Q1)is

not separable.

The solution to this problem is sought in the form:

S

(p1q

(QdQ+S

(?1q

$

C0(x

0 c01x

)'

-(Q t) dQdt.

The

substitution.

of (2.O.in the boundary conditions

yie1ds the condition:

(.-"lw

S

(?q

-t')4.QcLt

1-0 C1-0uC i-.

1Q1)[

i4'(?1q

(

t

C0(u) -

Q1I)i'(?1q)d.Q

0,

c0(.x) C01u)

+or z(?O

which corresponds directly to the condition (1.5.9.) for the

separable case. It follows that the influence functions and

f(PQt.-t)

can be specified by the following boundary problems:

in D, at x for

(z)

041-

p(1 Q

4

-f-or P6

C0Cx)17O,

(Vqj'o ç.or z(?)O,irti

0,

%.f'

7-(=O

I/Vt. 1)

with the initltal conditions:

(zo.1)

c.p(P1Qy=O

1

for z , in D, for 0 (z.D)

for z

(?0

,

in D, for ,O.

in the equation

(2Z.D.)'k., S1P1Q)

denotes spacial delta function on the contour C0(L

The remaining part of the potential can now be

ex-pressed as:

T'

with:

"1(Q .f

-(.tTx')(NL1-(Q 1b) t

*Nj')+

-

1

3ha']

,

foe-

:= z1,'.

f:?_

. ij 1D-

mD, at x

[(] w-(Q,

on

c0(11:),

The similarity between the equations (25.D.) and (i5.D.) should be noticed.

For the remaining potentials and the exciting terms on the free surface are again of nonseparable type, hence the problems defining these potentials can be represented as:

on z = 0, in D, at x

and it is convenient to look for the solution of the form :

1-

-('1t((?1Q'3O)d.q'

+

r(q',

Oct't

Substitution into (26.D.)b. produces the condition on the free surface:

Sh

Q'ItE

(('PiQ'j1-t):aq'cct t

(28,b

0

*[k+]q/(Pt)=

foi..

_Q

Hence the solution of the boundary problems (26.D.) is de-termined by the equation in which:

it

for

(.b)

and:

(

1rqL?1b)= 0,

in D, at x (3O.-D

b) [t

Qt(QL;s)

for z(q) = 0, in D,at x,

on C0(LX'

It follows that the convolution term in (2.D.) drops out.

On the :basis of the equations (2.D.), (29..D.) and (o.D.)

the potential can be. expressed as:

4

(34. D)

where the potentials correspond totthe following exciting

terms in (3.D' b:

=

-

') ()=

,

(z)

on z = 0 in D for.i = 2,3,4.

Simiiarly;the

t tial jcari:bewritten in-the-form:

.

Cv-C

with the exciting terms determining the potentials on the

right hand side defined as:.

-

L(E +

t

on z = 0, in D, i,j = 2,3,4.

Hence, finaliyi'thé potentials def-inedby -the:tboundary problems (1.4.30.) and (1.4.31.) can be expressed in the- corresponding

forms:

I

1

tlJj=.Z

From the above considerations it follows that:

the potentiaL describes the flow resulting from the steady forward motion of the ship.

the potentials describe the part

of the flow induced by the oscillations of the ship, which depends linearly on the magnitudes and speeds of the

oscillations.

-the twice indexed potentials ₎ j0i1;2.

describe the linear flow effects resulting from the inter-action of the forward speed with the ship oscillations,

the reference configuration, the terms resulting from the interaction with on :tI- ship hull, and the

terms following from the interaction with

on the. free surface..

d) the threeply indexed potentials describe the flow effects dependent nonlinearly on the ship oscillations, with

L.O , corresponding to the direct interaction

of roll with other modes, whereas and

-i,j

= 2,3,4,follow fromthe interactions on the hull

sur-face and free sursur-face respectively.

From the corresponding boundary conditions it is found that for the hull surface which. in the reference configuration is symmetric with respect to the plane y = 0, the following po-tentials are even functions of y:

toi

c211

_i2At.

_{fAt2..I'1 I}

whereas all reamining potentials are odd functions of y.

The equations (3.D.) can be rewritten in their equivalent

nondimensional forms:

S..-

(3D

't-

.2LL.

(YLSher

L2

in which:

o=

and:

for i

=2,3,4, j = 0,1,2,

whereas other potentials are normalized according to the

relations

(I. 3.21.).

Clearly the normalized potentials preserve the symmetry properties of their equivalents.

whence (i.E.) can be rewritten as:

.fr

4'Lt +[t-

Uti c&t

(a)

It' follows from the equations (6.D.)., (8..D.) and (15.D.) that

for the. ship performing harmonic oscillations of the circular frequency'(.J, the exciting term .rLin (3.E.) takes, the form:

=

Re[ ;:- e(t')]

for s. and potentials

pressionc'(4.E.) Re denotes the real brackets, L.

'P?

, andLrdenotes a

By inserting (4.E.) into (3.E.) and integration, it is found that:

R[ r eL&) L.,3')1

with:

t..rtt'O,

-o)

Appendix E.

Potentials of Flow Velocity for I-armonic Oscillations.

It has been shown in the Appendix D that for arb'itrary oscilla-tion of the ship with respect to the reference configuraoscilla-tion the potentials , I = 1,2 ...,4; , i = 2,3;

I = 2,3,4; are expressible in the form:

1.E')

compare with (2.D.). The convoltulon integral on the right-hand side of (i.E.) has been written under the assumption

that:

listed above. In the

ex-part of t'he term in the complex amplitude.

changing the variable of

L)

Since, according to (i.E.), (f?(t) represents the response of

and the principle of causality requires that there can be no reaction of the system prior to the excitation,, it follows that:

fo

<o.

Hence (6.E.) can also be written as:

cth,

(&E)

and the application of the inverse Fourier transformyie1ds:

DO

fiE=

(9.E')

The relation (6.E.) implies that can also be written in the form:

'1

I') .

A(L

(L)

(4o.E

where:

(Lt)t-

clk1

and it follows that:

By inserting (1O.E.), (11.E.'), (l2.E.) into 19.E.) it is found

that:

pe)

=

S{ tA(-]

t)

-

_3, )

besides the condition (7.E.) yields:

DO

I

0

0