The influence of water depth on the heaving, swaying and rolling motions and the bending moments of ships in regular waves

CHALMERS UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF NAVAL ARCHITECTURE

AND MARINE ENGINEERING

GOTHENBURG - SWEDEN

THE. INFLUENCE OF WATER DEPTH ON THE

HEAVING, SWAYINGI ROLLING MOTIONS AND

ON THE. BENDING MOMENTS OF SHIPS IN

REGULAR WAVES

by

C. H. KIM

DIVISION OF SHIP HYÔROMECHANICS REPORT NO. 43B-45B

Hydrodynamic Forces and Moments for Heaving,

Swaying, and Rolling Cylinders on Water of

Finite Depth

By Cheung H. Kim'

The potentials for forced heaving, swaying, and rolling cylifldeison the calm-water surface

of deep water used by Grim and Tamura are extended to be applied for the oscillations

of cylinders in shallow water by Thorne's method. Firstly, the hydrodynarnic forces and

moments on oscillating Lewis cylinders are calculated by Grim s method, secondly, the

exciting forces and moments on a fixed cylinder in a transverse incident wave system aré

calculated according to the HaskindNewrnan method; and, finally, the influence of the shallow-water effect on the forces ¡s illustrated graphically and discussed.

Introduction

WE CONSIDER two problems: (a) the problem of forced

oscillations in heave sway and roll of a cylinder on the calm water surface of shallow water, and (b) the problem

of a fixed cylinderjaa transverse incident wave system of

shallow water. Our problems are to détermine the br

drodynamic forces and moments on the cylinder in both cases and to determine the influence of the shallow-water

éffect on the forces and moments. There are two reports

on the first problem, by YuUrsell [5 ]2 and Wang [81, 'aff Scientist, Davidson Laboratory, Stevens Institute f Tech-nology, Hoboken, N. J. This work was done at the Institute of Ship Hydromechanics, Chalmers University of Technology, Goth-enburg, Sweden.

2_{Numbers in brackets designated References at end of paper.}

Manuscript recéied at SNAME Headquarters May 1, 1968.

Revised manuscript received January 8, 1969.

- _- Nomenclature -

-a = -amplt.ude of oscill-ation A = wave amplitude ratio

A, = source intensity in complex num-ber

B n beath

C n added mass coefficient E = exciting force and moment

F = force

g gravity constant

h = t'ater depth

= waveamplitude

H = half beam draft ratio or subscript designating heave

i = or subscript designating

imaginary part Im = imaginary part of

I" = added moment of inertia = moment arm

on" = added mass

M = momènt n = integer

damping coefficient

subscript dedgnating real part or

+ y2

Re = real part of

subscript designating roll 8= subscript designating sway or

sec-tional area time

T= draft

U = velocity amplitude of sway V = velocity amplitude of heave

and two repórts on the second problem, by Haskind [6]

and Newman [7].

Firstly, the potentials of the forced oscillations on the calm water surface of deep watér used by Grim and Ta-mura [1,2] are extended to those of shallow water by Thorne's method [3]. There are two reports on suèh

extensions [4, 5].

Secondly, by making use of the potential of shallow-water oséillation, the hydrodynamic forces and rhoments are calculated.

Thirdly, the exciting forces and moments on a fixed cylinder in beam waves are calculated by using the pre ceding calculations for fOrced oscillations according to the HaskindNewman method [6,7]; alternatively, the exciting forces and moments can be calculated by Grim's method [1,2], and a comparison of the results of both methods shows that they are in good agreement.

x,y = rectangular coordinate

a = pha.selagorlead

fi = polar coordinate tan-' (r/y) or fullness coefficient of cylinder

section

-ô = damping parameter = wave elévation

= deep-water wave number (c,2/g)

p0 = shallow-water wave number p = density of water

ça, 4' = velòcity potential

t n stream function

w = circular frequency

= angular velocity amplitude of roll. = subscript designating infinite

depth

Finally, sorne calculated results are represented in conventional dimensionless forin including the depth parameter as functions of dimensionless frequency. Extension of Potentials

First of all, we define our rectangular coordinate system

O-xy: the x-axis, on the calm water level, is pointing to the right and the y-axis, in the centerline of the cylinder at rest, is pointing downward. In the following discus-sion the irrotational motion of iñcompréssible inviscid fluid and the linearized boundary conditions and linear hydrodynamic pressure are assumed.

To begin with, the potentials for oscillations in deep water used by the authors [1] and [2] must be

repro-duced. They are generally represented in thé following

form

rvi

+ i'4'.

= I U Ie' A(ç',,c. + j'no,) (1)

Lcd

n=O

where V, U, = linear velocity amplitude of heave and sway and angular velocity amplitude of roll; A = com-plex source intensities to be determined by the boundary

condition on the cylinder surface;

_{+ i ç7j =}

partial velocity potential which satisfies the law of

con-tinuity everywhere iii the liquid, the free surface condition

at .y O, the condition of rest of Water at y and

radiation conditioñ at 1x1 -

;

= + ii&, =

partial stream function.

The partial potentials are for heave:

SOr + iço ₌

O kv

COSkx cik

- i1re cos Pz

.Çnr ± Pni

(l)n[cos2nß

+

_{(2n 1)}

'

cos(2n-1)ß

r2'

+iO

añd for sway and roll:

Or +

+

kp

sin kx dk

- i7rve' sin vX

-

r

vsin2nß

Çnro + iÇOnfto = r2 sin (2n + 1)81

±

J J where y = w2/g, r = /x2 + y2, 8 = tanLl(x/y)

e(z+)

and the wave source integral

k '

dk expanded

by Grim may be seen in [9].

The foregoing potentials for infinitely deep water are extended to those of limited water depth by making use of Thorne's method [3,4,51. We set the complete po-tential for the oscillations in shallow water, which are to be eìtended, in the following form.

+ i

=

[u et

Afl(h ±

(4)

Lcd

where the subscript h dêsgnates the potential for limited

depth of-water and =

+

imist satisfy the

following conditions:

Pnfl = O everywhere in the liquid:

V(4 ± - 'Pah = O òy ò at y = O; ò4Pnh .. O

at y = h;

òy

(4)

lim/ (" + vo4n)

= o,

x =

' = y tanh

voh, where y0 shalloW-watef wave number.

Since the potential p, does not satisfy the bottom

éondition (3), it is to be extended, by introducing an ad-justing potential , to tle corresponding potential of finite water depth, i.e.

Çah = ('pnrc, + ÇOn,.ad.) + Pib

The adjusting potentials ad for heave and sway and

roll are assumed in the following fòrms respectively.

For heave:

nrad.

f [a(k) sinh ky + ß(k)

-.

-X cosh k(h yi cos ¡cx dk

For sway and roll:

nrad.

=

f [(k) sitih ky + ö(k) cosh k(h - y)]

X sin ¡cx dk

where h

depth of water.; a(k), ß(k), 7,(k), ô(k)

unknown constants.

Firstly, the unknown constants ß(k) and &(k) are

eliminated by satisfying the free-surface Colidition (2) by

the potentials pnrad. alone.. Secondly, the unknown

constants a(k) and y(k) are determined by fulfilling the bottom condition (3) by the potentials (ç,

+

ad.).

Thus we deterimne the required adjusting potential

ÇCnrad. and the potential in the following forms: For heave:

-. e"

sinn icy - ic costiIcy

o _{¡c y p cosh kh k sinh kh}

X cos ¡cx dk (5)

'POr ad.

V

k°("')(k

v)ek

( 1)

2n-1)!

0

(5)

zì smb ky k cosh ky cont'd

y cosh kh k sinh khcos kx dk

cosh k(h y)

cosh kh - k sinh khcos kx dk

(2m-1)!

,f k2' (z,2

k2)

cosh k(h y)

ycosh kh - k smb khcos kx dk

For sway and roll:

IC y sinh ky - k cosh ky

o k-

z cosh kh - k sinh kh X sin kx dk Pn7 _{ad. = ,$1'} k2'1(k + v)e_k2 u sinh ky k cosh ky X . -

sinkxdk

u cosh kh k sinh kh

j°'

k cosh k(h sin kx cik ÇCOrh

,3o p cosh kh k sinh kh

-

k2''(u°

k2)

cosh k(h - y)

X .

sinkxdk

u cosh kh k sinh kh

FinaUy, the potential ço,, is determined by satisfying

the radiation condition (4). Before fulfilling the condi-tion, we need the asymptotic expression of the

poten-tial Pnrh for x

- ±

For this purpose we consider the complex-contour inte-gral of the function of the form F(k)e, which may

gen-erally represent the potentials in k-complex plane

As the contour we take the quadrant Re(k) > O, Irn(k) > 0, which passes over the singular point k = v0(0> 0),,

voi

the only positive root of the equation z' cosh v0h

-o sinh p-oh = O.

Taking the radius of the quadrant infinitely large and the indentation radius infinitely small, we find:

For hèave:

= Re[iriRes(vo)] For sway and roll:

cc1,7,, = Im[n-i Res(vo)]

By limiting process, the residues of the functions F(k)e

at k

v, i.e. Res (u,,), are calculated, and thus the

as-ymptotic expressions for z - ± are

For heave:

27r cosh2 voh cosh t',,(h y)

ccOrh

2v,,h + sinh 2z,0h cosh u,,h son VOXj

(_1)"

çcnr,,

(2m 1)! uo271(l - tanh2 v,,h) (9)

2r cosh2 cosh vo(h - y)

X

2poh + sinh 2u0h cosh v,,h vojx

For sway and roll:

2r eosh2 v1,h cosh vn(h - n) i ccor ± PO - --s 2v,,h + sinh 20h cosh z,,h X cos vo(xj

±

i

po2n +I(j,,fo2 v0h 1) (2n)!

2r cosh2 p0h cosh vo(h

- y)

X . _{cos voJxi}

2voh + sinh 2v0h cosh i'0h

We now returñ to the radiation condition (4) and sub-stitute the foregoing potentials in the condition and thus

obtain the potential Çnjh in the following forms:

For heave:

27r cosh° u0h cosh v0(h y)

ccoiiz

2y0h + smb 2,,h cosh u0h

X Cós voixi

in

_{vo2"(i - tanh2 z',,h)}

(2n-1)!

2ir cosh2 ,,h cosh u,,(h

-2poh + sinh 2u0h- cosh v0h

For sway and roll:

27r cosh2 h ccoo,

= F

t'02p,,h + sinh 2p0h cosh vo(h _y) cosh voh X Sin v0X =

i

z,02fl+1(aij2 1) (2n)!

2ir cosh2 ,oh cosh vo(h

sin 20h + sinh 2v0h cosh ,,h where corresponds to x -'. ± xi (10) (12) JUNE 1969 139 Po xi nr ad. n Tu

The extension is now complete. The two sets of the equations (6) and (11) and (8) and (12) are the extended potentials for heave and sway or roll respectively.

Since the adjusting potentials ço ad.( 'l'nr ad.) are obtained

in the form of Cauchy integrals, they are expanded in. series to be used for numerical calculation [5] (see Appen-dix).

Hydrodynamic Forces and Moments

The condition of zero relative normal velocity to be

sat-isfied on the cylinder surface for heave, sway, and roll may be written in the following form

Re [e" AnV'nh]

=

j

wt (13) n O

(X2 + y2)

where the stream functions A nI'n? depend où the mode

n=O

of motion. This condition determines the unknown source intensities and consequently the velocity potential

I, designated in advance in equation (4).

Having obtained the velocity potential ; one may

integrate the hydrodynamic pressure distributiòn on the submerged cylinder surface and deter-mine the hydrodynamic forces and moments:

[vertical forces in heaving

horizontal forces in swaying and rolling Llongitudinal moments in swaying and rolling

]

{Re (Ê

A f nh

s

])sin

xdx+ydy

+Im(Ê

Af

])coswt}

(14) n 0 8

zdx+ydy

where the coñstants An depend on each mode of' motioñ.

The forces (moments) consist of inertial and damping parts, both of which are in phase with the acceleration

and velocity of oscillation. They are called

respec-tively hydrodynamic inertial forces (moments) and hy-drodynamic damping forces (moments).

From the inertial and damping forces (moments) we define the added mass and added moment of inertia and damping force (moment) coefficients. These are

repre-sented in dimensionless forms. In case of sway and roll,

it is necessary, in addition to the foregoing, to define the added moment arm and damping moment arm, which re-late the moment and force of the inertial and damping parts respectively. For convenience the above men-tioned may be represented in tabular form, Table 1.

It is customary to represent the damping force and moment coefficientsNN, N5, and NR by wave amplitude

ratios Aif, A8, and AR, which will be discussed in the fol-lowing section. 140 inertial force damping force inertiál moment damping moment added mass added moment of inertia damping force coeffiëient damping moment coefficient added mass coefficient added moment of inertia coefficient damping force parameter damping moment parameter added moment 15, -' MSr arm

_{T - Fs,T}

damping moment 1SI

arm _F51T

LR

where

H'

2rcoshv0h o 'o 2v0h + sinh 2voh

°(th2 voh - 1)Ho'

(2n)! NR ÖR pwT ¿R, MR, T - FR,T ¿RI M51 - - F51T

Damping and Exciting Forces and Moments

'The asymptotic expressions of the velocity potentials cFh (z -+ ± ) for heave; sway, and roll are easily obtained by combining equations (9) and (11) and (10) and (12).

They are, omitting the subscript h, as follows: For heave

AH

cosh ,ìo(h

- y) e«+

cosh voh (15) where 9.,r 'sh 2voh

o-

_{2v0h + sinh 2v0h} i n

H vo2'(l tanh2 voh)Ho

(2n - 1)!

For sway and roll:

±

[Ul Ê AH'

cosh vo(h - _e«

Jn=O coshv0h

JOURNAL OF SHIP RESEARCH

Table 1 (8 = Beam, T = Draft)

Heave Sway Roll

PH, FHI F51 FR, FRI M5, M51 M5, Fil, m ,, F5, WV C., j,,

M,

FHI F51 H U

N5=

M51 mH' rna" = pB2 8

_PT

CR =

'r

pT

NH 55

-pwB2 pc4T2 xi +wt) (16)'

and

±:

corresponds to x -'- ±.

A : source intensities, depending on sway and roll The waves generated by the oscillations progressing at

far distance from the body 1ò'f'

= - -

at y = 0

g ot

are obtained from the foregoing asymptotic expressions in complex numbers, i.e.

For heave:

71H = vag

AnHne0t)

nO

For sway and roll:

Aif =

n=o

= ¡1 _A,ZHII'

no

n=o

AH'

/B/2

where A depend on the mode of motion.

In addii ion, by equating the mean power dissipated by the damping force in the forced oscillation and the mean power used in forming the progressive wave system in both positive and negative directions of the x-axis, we obtain the relation between damping coefficient N and

wave amplitude ratiò A in the following form:

Next we consider the vertical and horizontal exciting forces and longitudinal exciting moment on a fixed

cylin-der in a transverse incident wave system. Haskind's

theory [6 J stated that the exciting forces and moments

depend only on the asymptotic behaviOr of the potential for forced oscillations of the body with unit velocity am-plitude. Newman [7 J extended the theory and further es-tablished the formulae relating between exciting forces and moments and wave amplitude ratios A in the

two-dimensional case. The exciting free (moment) in our

case is represented by the following contour integral [7]

E = iwpe I I

c'i -

-

I dy (20)

J0L

òx

where

c'i potential of transverse wave system

= asymptotic potential of unit velocity amplitude

We take two incident waves of the following form, each of which is oncoming from the positive and negative ends of the x-axis:

(17) .gh cosh vo(h _y)

Çj = i

w cosh v0h

Each asymptotic potential c' is obtained from the po-tentials H, ,and Rby taking unit velocity amplitude

[see equations (15) and (16)]. Substituting the above defined potentials in equation (20) and executing the in-tegral we obtain the heave-exciting and sway-exciting forces, EH and E5, and the roll-exciting moment ER:

[EH

lE5 =

LER pghet A,,H,, n=o

±i >A,H'

n=o

±i

AI,H,,' n=O ¡EHI

'(i±

2v0h pghB poB\ sinh 2v0h lEsi A5

₍

+

2voh

pgvohS vo2S\ sinh 2v0h

IERI ARH(1

+

2v0h

pgvokST vo2S sinh 2v0h

where

S = submerged section area H half beam thaft ratio

T = draft

The phase differences between the exciting fòrces (moments) and both oncoming waves from right and left are obtained by comparing the phase angles of the wave

2voh + sinh 20h

2 cosh2 0h

(21)

(22)

where ± correspond to the waves which are oncoming

(18) from the positive and negative ends of the x-axis; A

depend on the mode of motion. Making use of the rela-tion = V tanh v0h arid the formulá of the wave ampli-tude ratios (18), we obtain the magniampli-tude of E in

cOnven-tional dimensionless forms:

(23) JUNE 1969 141 NH N5 NR = 4H2 A AR2(B/2)2 WZ'O \

±

).

(19) sinh 2v0h

iR =

iPaR/B/2 A,H,,'e« 'oxI +cL)

n=O

where

aif, a5 = linear amplitude of heave and sway

= corresponds to z

-Consequently, the wave amplitude ratios for heave, sway, and rollare written in the following form:

'is =

ivas AnHnFe«_v0I+t) nO

r

i

i

elevations Rel -- p,e"

with those of exciting forces

L g Jy=O

(moments) Re [EH,ES,ER]. These are for

ag: while for sway and roll they are a8 aR ± (ir/2), where

=

ar(AnHn)

[asl

=

arg(

AH')

(25)

LaRJ

\,,°

/

It should be noted that the minus sign of phase difference

means the phase lag and vice versa. The value of A,,

for sway and the value of Anfor roll should be different,

but the numerical computation revealed that, for our low-frequency range, a8 = aR. This is because the wave

pressure distribution on a fixed cylinder consists of

sym-metrical and asymsym-metrical parts with respect to the

y-axis: the symmetrical part contributes to heave-excit-ing force, and the asymmetrical part contributes to both swar- and roll-exciting forces; and both of the latter take the maximum values at the same instant for the low-fre-quency range. However, it should be noted that, since the y-axis is positive downward and the positive sense of moment is chosen to follow the right-handed system, then, when the sway-exciting force becomes positive maximum, the roll-exciting moment takes a negative maximum value. It is sufficient, therefore, to calculate heave- and sway-ethting forces together with the roll-eÑciting m9ment arm.

lE

-

_ARH

T - A8

Numerical CalculatiOn and Discussion

By the transformatioiÇx + iy

° +

ae

+ be°)

t.he stream functions in tliêboundaryconditiort[&quatiii (13)1 ate represented as functions of three variables O, a, and b. The funCtions are expanded in finite trigono-metric series on the cylinder contour, namely in even sine

series for heave:

/'fl0(a,b,e) =

bo(

-

e)

± E b,,,,,

sin 2mO andin odd sine series for sway and roll:

- 4

Iflh(a,b,O)

= E

Cnm sin (2m. - 1)0

m=i

--The boundary conditions are thus reduced to ten and eight. simultaneous linear equation systems respectively,

from which the source intensities are determined.

The adjusting potentials or stream functions are given in infinite trigonometric series (in the appendix), and in our calculation the first nine terms were taken. In the numerical integration of the singular functions G2+1(vh) and F23+j(vh) the limits were taken at 2p = O and 30 and the increment of (2p) was 0.5 (see Fig.. 2). . Thç,

calcu-lated forces and moments are represented as the

func-(26)

142

tions of dimensionless frequency in the figures. The

numerical calculations were carried out by the IBM 360 computer at Chalmers University.

Our discussion will be primarily concerned with the influence of the finite-depth effect upon the hydrodynamic behavior of forces and moments in the very low fre-quency range for both heave and sway.

Wave Length

To begin with, we consider the lengths of the waves generated by the oscillations on the surface. of a limited

water depth.

The formula for shallow water wave

number, o' = v tanh 0h, reveals that in the low frequency range the relation Po> o' is valid. This explains that at the same frequency of oscillation the generated wave length of shallow water is shorter than that of deep water. It should be noted that the approximate relation ve/o'

1/(v0h) is valid at o' -'- O (Fig. 1).

Source Intensities

In our discussion we are obliged to depend on the cal-culated source strengths. For vanishing frequency o' - O

we need to consider only the significant source intensity

A0. From Fig. 3 we see that firstly at o' = O the

non-dimensional source intensities for heave Ao7ir/B and Aor/B approach Unit and zero respectively for any depth

Of \vater; secondly, in the vicinity of o' 0, the intensity

A017r/B for finite depth becoms higher than that of

in-finite dept.h as the depth decreases, while Ao77r/B for in-finite

dept.h decreases first and then increases as o' increases. Although Fig.. 3 shows the calcflated result of a circular cylinder, t.he foregoing -statemnts are valid for any form

of cylinder. As to the swaying oscillation, it is known

from several calculations that the intensities A for finite dept.h are generally higher thkn those of infinite depth when o' - 0.

Added Mass Coefficients

One may obtain the approximate formula of the added mass coefficient for heaving, oscillation of a circular cylinder in t.he following form

CHnnCH+G28+l(o'h)

1i

[- ln(!)

-

0.577]

From the above formulae we see that the influence of the shallow-water effect on tle behavior of added mass coefficients as o' -- O may be eplained by the behavior of

the singular fünction G21(o'h) I (see Figs. 4 5, 6, and 7

80

in connect-ion with 2).

It is stated that CH for finite

depth is- higher than that of infinite depth of water at o'

O and this i_s valid for any forin of cylindér. Let us

ob-serve for a moment the added mass coefficient at -high

frequency. Figs. 5 and 7 illüstrate the added mass

co-JOURNAL OF SHIP RESEARCH

heave always

± (ir/2) and

A 40 30 2.0 1.8 1.6 14 1.2 1.0 08 0.6 04 02 0.0

r,

vh = v0h.tanh v0h

I

I

I,

t.

V4

.I

..r

_I1

rira

r

vh Fig. i

Shallow-water wave number p0as function of frequency vand depth h

1.5

G2i (vh)

Fig. 2

Curves of the singular integral G2.1 (rh)

O 0 0 2 0.4 0.6 0.E 1.0 1.2 1.4 1.6 1.8 2.0 3.0 vh o 2 5

i i 1.0

09

0.8 7 0.6 p.5 0.41 0.3 0.2 0.1

Tii

100

h 1.3 1.5 2.0 - Ure11 0.0 i . i O 0 0.2 0.4 0.6

08

1.0 H =1 p = 0.735 1.2 1.4 'I H = 1 A

'

A ' or oi = O.75 1. O

Fig. 4 Added mass coefficient for heave

Fig. 3

A UI 1.2 i .ot 0.8 0 6 0.4 0 2' 0.0'

00

1.0 2.0 3 o Yu - Ursell h .0

₄₀

H'= i = 0.785 5.0 = 1.5 2.0 4.0 _10.0 B 6.0 CH 1.8

16

14 1.2 1.0 0.8

06

0 4' 1.75 2.0 4.0 _{10.0 .OD} 11=1 Fig. 5

Added mass coefficient for heave

Fig. 6 Added mass coefficient for heave

0H 1.6 1.4 0.2 vB 2 O 0 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4

H _2.0 O 4 0.2L 0. 0

00

= 1.75 = 1 1.0 2.0 3.0 4.0 vB 5oo 6.0 3 2.5 2.0 1.5 1.0 0.5 0.0 H: = i

31

10 6Q

15

0 0 0.2 0 4 0.6 0.8 i.O 1 2 Fig. 8

Added mass coefficient.for sway

Fig. 7

efficients CH at a wide frequency range. We see from them that at high frequency the values of C11 for finite depth are generally higher than those of deep water. and from Fig. 5 that the values of CH tend to approach the known values of CH for i'

,

as given by Havelock

[11]; i.e.

1-

1fT\

CHCHIl+

-ih

h-+coL

It appears that at infintely high frequency the value of CH by Yu-Ursell tends to be higher than that given by

Havelock (CH = 1.125), while the author's C11

ap-parently approaches that by Havelock. It. seems that

Yu-Ursell's numerical calculation might not he accurate enough in comparison with the author's.

Fig. 7 also illustrates the tendency of CH in the very

high frequency range, which seems reasonable. Now we

corne to the discussion on the added mass coefficient C5 in swaying oscillation. For infinite depth of water at

p 'O the exact formula of C5 is

C

_(1 --a)2+3b2

_(1a+b

This value is also obtained numerically by applying the potential of infinite water depth at the vanishing fre-quency-. i.e., A07sin ß/r.

For the finite depth of water at ' -+ . on the other

hand, the velocity potential is approximately written by

G23+i(vh)j

_r

_sinß]

and since the source intensity A07 for finite vater depth is

in general relatively higher than that of infinite depth and

since we see the behavior of the singular function

G2sj(vh).l,i at y O (see Fig.. 2), we come to the

conclu-sion that C5 for finite depth is higher than that of infinite

depth when y -- O (see Fig. 8) and the behavior of C.5 in

the vicinity of y + O is caused by the cont.ribut joli of the singular function G28+i(vh)Ie...,i.

Wave Amplitude Ratio

The influence of the shallow-water effect on the wave amplitude ratio A11 and A5 at y O may be easily dise

cussed by observing the asymptotic formula for ' -+ 0,

namely, for heave:

fvB\

_-

f

i\

fvB

A11 =21---,;A11=--jj

h.o'

\

/

hoh \'Ot/_..O

and for sway:

-

4.

/

---fpfi\2

V¿'Or T L10 _s

o D

= o2

+

.

()

The slopes of the curves A11 and A11 at the origin are

re-h.o

h=h

r h2

spectively 2 and (-). The value of the latter is

in-fini tely large (Fig. 1). It is obvious therefore that the

amplitude ratio for shallow water is higher than that of deep water at 0 (Figs. 10,11, 12).. Further, it is stated that the that of amplitude ratio for shallow waterlbhigher

than deep water throughout the whole frequency rarge

(Fig. 11). This is due to the contribution of the

higher-order potentials to forming the progressive wave system

[equation (18)].

As to the amplitude ratio for sway, it is easily observed from the above formulae that, at y 0, A5 is a

second-o

order parabola with zero slope, while A5 is a straight line hoh

with finite positive slope; thus both curves must cross each other (Figs. 1-3, 14). It should be noted that t.he

coefficients for roll, CR and AR, must-have similar

charac-ters to those of sway, for both oscillations are essentiallr similar from the point of view of antisymmetry about the

y-axis (Figs. 9, 14).

Added Moment and Damping Moment Arm

The added moment arms 157/T and 157/T for shallow water are generally shorter than those of deep water at y - 0, but at the other low-frequncy range in the vicinity

of y O the opposite phenomena appear (Figs. 15,16). The damping moment arms l5/T and lRj/T and the

ex-citing moment arm 15/T coincide with each other and de-crease as the depth of water dede-creases .(Fig. 17).

Exciting Forces

n the preceding section the discussioû was confined to

the HaskindNewman method. We can also easily

calculate the exciting forces and moments by Grim's method. By applying thé latter method the same calcu-lations were carried out and both results were compared.. it was found that they were in good agreement. In this report the results by the. HaskindNewman method were represented in Figs. 18, 19, and 20. It is obvious from the wave amplitude ratios A11 (Figs. 1OE-14) that the heave exciting forces of shallow watér are generally

higher than those of deep water.

Similarly, the ex-planation of the behavior of the sway-exciting force may be obtained from the behavior of the wave amplitude ratio for sway, A.8 [equation (23)1. The exciting moment

arm 1E/T is represented in Fig. 17.

Minimum Depth

To find the minimum allowable depth parameter h/T

for which our solution converges, the calculation has been

carried out for several widely varying forms of cylinders. From the comparison of the results it vas concluded that:

i

For deep-draft cylinders the rmmmum parameters are very low.

2 For shallow-draft cylinders the values of minimum parameter are relatively high.

3 The influence of fullness coefficient ß on the

Fig. 9

Added moment Of inertia for roll

H= 1 = i 0 0 0.2 0.4 0.6, 0.8 1.0 1.2

-00

02 0.4 0.6. 0.;8 Yu - Ursell 3.7,8 1.0

Fig. 10 Wave amplitude ratio for heave

p = 0.785 theory h

.

= 1.75 experiment 0.3 1-t vB

z

m Yu - Ursell h = 2.0 1.0 0.9] 0.8_ O. 7 _{0.6= 0.5} 0.4 0.3 0.2 O il 0.0 10.0 h= 1.5 2.0 4.0 I I j

--0

'o H i H= i ß = 0.785 AH

Fig. 11 Wave amplitude ratio for heave..

Fig. 12

Wave amplitude ratio, for heave

0.0

02

0.4 0.6 0.8 1.0 1.2 1.4

00

0.2 0.4 0.6 0.8 1. O

Fig. 13 Wave amplitude ratio for sway

H= i =

1

1.2

Fig. 14 Wave amplitude ratio for roll

1.6 1 4 1.2 1.0 0.8

06

04

t- o C

z

0.2 I- o

'I

0 O

G

z

in

0

'o 0.50 O 45 0.40 0.35 0.3 0.25 0.20 0.15 0.10

A

A

0.0 = 10.O,cD 6.0 4.0 2.0 0.2 0.4 0.6 0.8 1.0 1.2 vB 'Rr T 1.1 o.q o. 0.7. 0.6 0.5 0.4 0.3 0.2

00

14

AVA

-'r

0.2 0.4

10.0.

0.6 0.0 H = 1 1.0 1.2

Figé 15 Added moment arm for sway

Fig. 16 Added moment arm for roll

131

1i

1

--

----0=1 go 80 70 60 50. 40 30, 20

Fig. 17 Damping moment arm for sway and roll and exciting moment arm

Fig. 18

Heave exciting force coefficient

00 0.2 0.4 0.6 0.R 1.0 1.2 H=1 0.785

pgB

t- C

z

m 'o o' 'o

10

0.9 0.8

07

0.6 0.5 90 80.0.4 70 - 60_0. 3 50 40 0 2 30 20 .0.1 10_

000

0.0

0?

0 4 0.6 0.8 1.0 Fig. 19

Heave exciting force coefficient

11=1 1.2

1.4

mum allowable depth is also significant when the shallow-draft cylinders are considered.

4 The minimiïm parameters for sway and roll are

lower than those of heave.

Conclusion

In general, the influence of the shallow-water effect on the hydrodynamic forces and moments is remarkable. The curve of the added mass coefficient C(CH, C8, CR)

for finite depth generally crosses the curvé of C for infinite

depth twice, and thus the twö crossing points divide the

frequency range into three parts. The values of

are first higher, then lower, and then higher again than

those of Ch...,. at the first and second and third frequency

ranges respectively. As to the wave damping or wave

amplitude ratio A, AÑ is generally higher than AH

h=h

_h.

throughout the whole frequency range, while, (A s or AR)hh is higher and then lower than (A8 or

at y O and in the vicinity of y

+

O respectively

Acknowledgments

The author expresses sincerely his gratitude to Pro-fessor Falkemo, the head of the Institute of Ship

Hydro-mechanics, who supported this work.

Above all, the author is deeply indebted to Professor Grim for furnishing his unpublished computer program for heave and pitch of ships in regular waves, and to Professor Porter, who allowed the author to copy the work of Yu and Ursell from his own library and willingly helped him by. sending the recent report on a heave theory of a sphere in shallow water by Dr. Wang.

This work has been financially supported by the Swed-ish Council for Applied Research.

References

1

0.

Grirn,

"Eine Methode für eine genauere

Berechnung der Tauch- und Stamph-bewegungen in glattem Wasser," H.S.V.A. Bericht No 1217, 1960.

2 K. Tamura, "The Calculation of Hydrodynamical Fôrces and Moments Acting on the Two-Dimensional

BodyAccording to the Grim's Theory," Jour. SZK,

No. 26, September 1963.

3 R. C. Thorne, "Multiple Expansions in Theory of

Surface Waves," Proc. Camb.. Philos. Soc., Vol. 69, 1953.

4

W. R. Porter, "Pressure Distributions, Added

Mass, and Damping Coefficient for Cylinders Oscillat-ing in a Free Surface," Report 82-16, Institute of En-gineering Research University of California, 1960.

5 Y. S. Yu and F. Ursell, "Surface Waves Generated

by an Oscillating Circular Cylinder on Water of Finite Depth:

Theory and Experiment;" Journal of Fluid

Mechanics, Vol. 11, 1961, pp. 529-551.

6 M. D. Haskind, "The Exciting Forces and

Wet-ting of Ships in Waves," DTMB Translation 307 by J. N.

Newman, November 1962.

7 J. N. Newman, "The Exciting Forces on Fixed

Bodies in Waves," JOURNAL OF SHIP RESEARCH, Vol. 6,

No. 3, December 1962, p. 10.

8 S. Wang, "The Hydrodynamic Forces and Pres-sure Distributions for. an Oscillating Sphere in a Fluid of

Finite Depth,"

Dissertation,

MIT, Cambridge,

Mass., June 1966.

9

C. H. Kim, "Uber den Einfluss nicht linearer

Effekte auf hydrodynamische Kräfte bei erzwungenen Tauchbewegungen prismatischer Körper," Shiffstechmk

73, Heft. September 1967.

.10 J. V. Wehausen, "Surface Waves," Handbuch der

Physik, Band IX, Springer Verlag, 1960.

11

T. H. Havelock, "Ship Vibrationsthe Virtual

Inertia of.a Spheroid in Shallow Water," INA, January

1953.

12 _{M. D. Haskind, "Waves Arising From Oséillatión}

of Bodies in Shallow Water," SNAME Technical and Research Bulletin No. 1-32, 1961.

13 Whittakér and WatsOn, "Modern Analysis," Ex. 2. 12.2.

Appendix

Adjusting Potentials, and Stream Functions for Heave

-

G23i(vh) ÇOrad. o (2s)! h2' (2e + 1)! G23±(vh) r28" cos (2s + 1)ß 'cos 2sß i ( .J..L r23

-

-Or ad. s=0 (2e

± 1)!

h2'

sin (2s + 1)ß

ÊG2(,i)i(Ph)

_{(2s + 2)! h232}

r2'2 sin (2s + 2)6

'Pnr ad.

(i)

Ê

F2+23...1(vh) (2n 1)! s=0 (2e)! h25+23 r2' cos 2s8

(_l)a

' 1 F2,+i,_i(vh) _.2+1

+

_{(2n I)! (2s + 1)!}

_h22'

X cos (2e + 1)ß (-1) . i

F2 i,2(h)

nr ad. (2n 1)!

L

(2s h272+2'

x r231

sin (2e + 1)ß ±

(n

1)!

L

(2s ± 2)!

F2231(vh)

r232sin (2s + 2)ß

Adjusting potentials and stream functions for Sway and roll Çøo, ad. = p ,o (2s+2)!h232

r2'2

sin (2e

± 2)ß

+

(2 -f 1)!h23+2 r281 sin (2e + 1)ß

ad. Çnrad. = =

v

G28+i(vh) r28 cos 2sß -o (2s)! h28 r23' cos (2s + 1)

+

. G283(vh) (2s + 1)! h282 p F2+23+i(vh) _r28+2 (2n)!

'o (2s + 2)! h2282

X

sin (2s + 2)ß +

sO

(2s +1)!h2t1282

X r28 sin (2s + 1)ß p ad.

= -

Ê

F223_j(h)

r28 COS2s/3 (2n)! =o (2s)! h228

+

₍₉₎

F28s3i(vh) r28+l cos (2s + 1)ß

-

. o(2s + 1)!h2282 _UU23+1 G23j(v) =

(u - u)(v :osh u - u

sinhu) du

-

U2'(U + v)e1

du

F28+i(v) _{= ,YO u cosh u - u sinh u}

The expansion of the foregoing Çaucy integrals for numerical calculations are obtained from Reference [5].

Abstract

The heaving and pitching motións of a Series 60 model of = 0.7 moving in longitudinal regular head waves of shal-)w water are calculated by Watanabe's strip method [1], [2], 3]. The results are represented in Figures and the shallow

'ater effect is discussed

IntroductiOn

By applying Watanabe's strip the'oiy sorne important

hydro-ynamic forces and moments acting on a Series 60 model of

0.7 moving in. a longitudinal head wave system of

shal-w shal-water are calculatd. The calóulated results are

represen-d in non-represen-dimensional forms anrepresen-d shown in Figs. A.

Subsequently, heaving and pitching motions of the ship are alculated and the results are presented in Figures B.

The influence of shallow water depth on the hydrodynamic arces and motions as shown in the figures are discussed.

It is revealed by the calculatións tht the heaving and pitch. ig motions are remarkably damped by the shallow water

ifect.

Definition of Ship Motióñs nd Waves

The coordinate systems here utilized are space- and

body-jordinate system 0--XYZ and G0-xyÈ respectively. X-axis lies n the undisturbed water suEface and. Z-axis points vertically

pward. x-axis is longitudinal passing through the center of

ravity G0 of the ship, while y- and z-axis point port and

Up-ard, respectively. The coordinate system Go-xyz coincides

ith the system 0XYZ at the initial rest conditiön. We follow e convention of right-handed coordinate system.

I z

z

The influence of Water Depth on the Heaving and.Pltching

Mótions of a Ship Moving in Longitudinal Regulär-Head Waves

Cheung H. Kim

Assuming only heaving and pitching motions of a ship the speed V in a longitudinally oncoming wave system, we Iscribe the surface wave aá follows

-= Ecos (v0x'± q)et), (1)

here

h wave amplitu,de

y0 shallow water wave number, i. e

(w2Ig y0 tanh v0h)

We circular frequency of èncounteri. e.

(w + v0V).

The heaving and pitching motions of the ship

correspond-ing to the wäve defined above are then expressed by

=

cos (0)et ±

W = Wac0S (wt-+ e)

respectively, where era' are-heave -and pitch amplitudes

and e, a

phase differenceshètween- heave and wave and

pitch and wave, respectively.

The Coa pled Equations-and Coefficients

The coupled equations of heaveand. pitch of a ship movmg in longitudinal regular waves [li, . [2-].- are written in the form

a, + b + cdie4gi4,. =SFa.cos (Wet + £FW)

H

The coefficients on the left-hand sides of the above

equa-tions are

a = V + m" dx

L

bSNdx

-L

c2gfydx

L d

Jm"idx + jSxdx

L L

e = JNxdxV$rn" dx

L L

g = 2gfyxdxVJNdx

L L A = I, +_$ m" x2 dx L B .fNx2dx -L

C=2gJyx2dx_VE

L

D =Jm"xdx + fSxdx

L L

E =$Nxdx + VJm"dx

L

L-G = 2gJyxdx

L

where water density . -.

g gravity constant

V displacement volume

S.

sectional area under calm water level

y hall-breadth of a section on the calm water-line longitudinal moment of inertia of the ship's

mass about y-axis

-m" sectional added mass of unit thickness for heave -N sectional heave damping coefficient of unit thickness -The exciting forces and moments on the right-hand sides of the equations (3) are represented in the form

Ic08eFWÌ coshv0 (hTm) Jcosv0E

Fa

(.2Qgh$y

- (dx

smEFw, L coshv0h . .-.tsinv0x j

-,, -cash V0 (h - Tm). f cos vx dx - wh (w + v0V) j' m L coshv0h ( sin vx J (2) - 127 - Scliiffstethnik Bd. 15-1968 Heft 79

Ma COS Eg L sin EI

T wh N

cosh V0 (h - Tm) I 5jfl V0X cosh v0h _i cos v0x and cosh v0h respectively. DimensionLess Representation

In representing the calculated results, the following dimen-sionless forms are-used:

h

depth -parameter

T

wavelength to ship length ratio L

-. - F

Froude number gL

we 1/L/gfrequency of encounter

- a

virtual mass coefficient b T/L g

QgV

A

virtual inertia coefficient V L2.

B V" _{pitch damping coefficient}

gVL2

-F - -.

exciting force coefficient

QgAh

Ma

exciting moment coefficient Q g TW'o h

heave damping coefficient

pitch amplitude ratio

J

dx

are replaced by y = w2/g and eTm,

RB

=

{

/ :'tI CO5 EpW - COS I

2h \ A

- 147L _- ¡ xL \SÌ12

sin LW - Sm -E,w - 51fl

2h \ A

wave length

length between perpendiculars waterplane area

moment of waterplane area about y-axis.

cos ew

- W0L

sin E + Sin LW - 2h Wa L

)

Calculation and Discussion

For the numerical calculations we adopt a Series 60 mod

In the calculation the following assumptions are made:

Although trim and parallel sinkage exist they are w

considered.

The center of gravity lies at L/2.

The Virtual Mass, VirtUal Inertia, Heave an

Pitch Damping Coefficients are represented -as fun: tions of frequency fòr different depth parameter h/T F11 0.0. They are illustrated in Figures A-1 to A-4. It

seen from Fig. A-1 that at high frequency range the addL mass for the depth h = 1.5 T is approximately twice as lar as that for h/T = Oc, 10.0, 4.0, and 2.5. This suggests th

the natural heaving period of a ship in shallow water will u

longer than that in deep water, provided that damping is co'

paratively small. From Fig. A-3 we obsèrve that heave dam ing coefficients increase noticeable as the depth decreases a

at the depth of hIT = 1.5 they are nearly twice as big as tho for h/T = 00 and 10.0.

The Exciting Forces and Moments acting on t

restrained ship moving at the velocity of F11 = 0.2 are ill strated in Fig. A-5 and A-6 as functions of wave length ship length ratio. It is found that these forces and morne

generally increase as the depth decreases.

-Station B (x) H(x) (x) 1 0.0830 8.2425 0.8386 3 02803 0.8186 0.8716 5 0.4001 1. 1685 0:9301 7 0.4290 1.2498 - 0.9761 9 0.4280 1.2498 0.9860 11 0.4280 1.2498 0.9850 13 0.4280 1.2498 09&13 15 . 0.4113 1.2010 - 0.8660 17 0.3372 0.9848 0.6794 19 0.1575 o-4599 0:3751 cosh y0 (h -h j (w2 m" ---2 Qgy) Tm) f cos v0x _}dx L coshv0h ( sin v0x h dm"" còsh ± (h - T) ( Sin V0X where J dx A 1i s L \ dx- / cosh v0h i cos v0x where L water depth _h A

Tm mean draft of a section w

F =

-A/L = hIT = 0.0,0.1 and 0.2. 0.4, 0.5, 0.6, 0.1, 0.8, 0.9, 1.0, and2.0. . 00, 10.0, 4.0, 2.5, 2.0 and 1.5. 1.1,

of CB 0.7 having following particulars.

Length between perpendiculars 3.000 m

Displacement volume 0.1537 m

Draft . - 0171 m

Beam...

0.428 m

Radius of gyration 0 750 m

Schiffstechnik Bd. 1968- Heft 79 - 1-28

8MW phase differences between exciting force and wave and exciting moment and wave, respectively.

Sectional values of added mass and damping coefficient m" and N for heave are obtained from [3]. In the case of deep

water these values are obtained from [8]. If h-+ o then y0

cosh y0 (h Tm)

h heave amplitude ratio

Wa h

By assuming that CG. lies at midship the absolute bow and -stern motions and relative bow motion are expressed in

non-dimensional forms as follows

AB

h _1/ r=a

-

Wa'-cos £w + - Cos LaW)

2h

h

where B (x) beam of a section

H (x) half-beam draft ratio of a section f3 (x) fullness coefficient of a section

The calculations are carried out for the following speed:

waves and depths.

1.3, 1.5, 1.7

-5-. o h _1.5

r'

4.0 A 10.0

1-.

p

1 0 1 5 2.0 2.5 3.0 3.5 40

Fig. A-2 Virtuel inertia Coefficieñt at F5 = 0.0

0.12 o_lo ò.0S 0.04.. 0.02_ 0.06 Series 60, ¿ 0.7 = 1.5 ô.00 . 10 1.5 2.0 2.-5 3.0 - 3.5 . 4.0

Fig. A-4 l3ìtch Damping Coefficient at F5 0.0 0.06.. 0.04_ O 02_ fi;-o.ôo t -

I..

-129-.

Seife. ì5,- 1968

- Heft.79 Serleö 60, C3 0.-7 p A Seriee 60, C,5 0.7 0. 20_ 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fig. A-1 Virtüel Mäss Coefficient at F5 0.0

Serles 60, CB 0.7

.1.0 1.5 - 2.0 2.5 3.0 3.5 . 4.0

Cp, 160 _140 120 .100 .80 60 - 40 - 20 o 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -0.0 2.5 4.0 10.0. Ma p g i B h _{= 1.5} 1o.0 -2.0 -Ser1e 60, CB 0.7 h 2.0 1.8 1.6 1.4 1.2 1.0 0 8 0.6 0.4 20 1.8 1.6 1.4 1.2 1.0 0 8 0 6

Fig. A-5 Exciting Force Coefficient on the Restrained Ship Fig A-6 Excitiñg Moment Coefficient on the Restrained S

Moving at F 0.20 Moving at F = 0.20

-0 6 08 1 0 1.2 1.4 1.6 1.8 2.0 0.6 0.8 1.0 1.2 1.4 1.6 18

Fig. B-1 Héave Amplitude Ratio at F = 0.0 Fig. B-2 Heave Amplitude Ratio at F0 = 0.10

-4

ti

u,'

o V h _{- 1.5} 2,0 4.0 10.0 -= 10.0,-I-. 5 2_o çcv Serbe 60, CB 0.7 4.0 4.0 10.0. -1.5 ßeriea 60. CB - 0.7 o -20_ -40 -6 -1 00L1 .0 -120 -1400.8 sp h _1.5 10.0. Seriee 60, CB 0.7 - 1.5

Fig. B-4 Pitch Amplitude Ratio at F = 0.0

Bert.. 60, C 0.7

0 6 0.8 1.0 1.2 1.4 1.6 1.8 2.Ò 1.0 1.5 2.0

Fig. B-5 Pitch Amplitude Ratio t F 9 0.1 Fig. B-6 Pitch Amplitude Ratio at F 0.2

- ]31 - Schiffstechnik Bd. 15- 1968- Heft 79

0.6 0.8 10 1.2 1.4 1.6 1.5 7.0

Fig. B-3- Heave Amplitude Ratio at F = 020

The Heave and Pitch Amplitudes together with the phase differences with respect to the waves are illustrated in

Figs. B. In general the motions are remarkably damped as the depth decreases. This tendency is more significant as the Froude Number increases.

Although our theory cannot solve the motion problem in very shallow water, it will be quite useful to consider under-keel clearences. Provided that this theoretical calculation is

proved to be reasonable in an experimental study, this method

will be a routine technique for further research on ship's

be-haviour in restricted waters.

-Acknowledgements

This research was carried out with financial support of

Chalmers University of Technology.

The author expresses sincerely his thanks to Prof. C. Falkemo, Head of the Division, fo-his constant support.

The numerical calculations were carried out by the

com-puter IBM 360 at Chalmers University.

(Received 5th July 1968)

- Nomenclature

a, b, e, d, e, g coefficients of heave equation A, B, C, D, E, G coefficients of pitch equation

A waterplane area

B (X) beam of a section Fa exciting force amplitude

g gravity constant G0 cênter of gravity (C. G.)

h water depth h wave amplitude H hall-beam draft ratio

moment of waterplane area

moment of Inertia of the ship about y-axis L length between perpendiculars

m" sectional added mass of unit thickness for heave

Ma exciting moment amplitude N sectional heave damping coefficient

SW sectional area under calm water surface

t time

T draft

Tm mean draft V ship velocity. V displacement volme-w suffix designating volme-wave

Sehiffstechnlk Bd. 15 - 1968 - Heft 79 ₁₃₂

-SCHIFFSTECHNIK

-- Forschungshefte für Schiffbau und Schlffsmaschinenbau

Verlag: Sthiffahrts-Verlag Hansa" C.Schroedter & Co., Hamburg 11, Stubbenhuk 10. Tel. Sa.-Nr. 364981.Schriftleitung: Prof. Dr.-Ing. Kurt Wendel, Hamburg. - Alle Zúschrilten sind an den obigen Verlag zu richten. - Unaufgefordert eingesandte Manusknpte werden nur auf ausdrucklichen Wunsch zurückgesandt - Nachdruck auch auszugsweise nur mit Genehmigung des Verlages. - Die SCJSIFFSTECHNIK erscheint fünfmal jährlich. Abonnementspreise: Inland jährlich DM 34,90 eInschl. Versand-kosten und Netto-Umsatzsteuer von 50/e (5'/: 0/, ab 1. 7. 68); Ausland: jährlich DM 36, eInschl. VersandVersand-kosten. Einzelpreis: DM 7,50 emschl Netto-Umsatzsteuer zuzüglich Versandkosten Abonnements Kündigungen müssen bis spatestens einen Monat vor Ablauf des Jahres-Abonnements beim Verlag vorliegen. - Anzeigenleitung: Irmgard Dahi, Hamburg. - Auzeigenpreisliste Nr. 4. - Der Auftraggeber von Anzeigen trägt die volle Verantwortung für den Inhalt der Anzeigen. Der Verlag lehnt jede Haftung ab. -Bankkonto Vereinsbank Abteilung Hafen - Postscheckkonto Hamburg Nr 141 87 - Höhere Gewalt entbindet den Verlag von

jeder Lieferungsverpflichtung. - Eftüllungsêrt und Gerichtsstand Hamburg. - Druck: Sehroedter & Hauer, Hamburg 1. x, y, z body coordinates

X, Y, Z space coordinates section fullness coefficient

eCW phase difference between heave and wave

e phase difference between pitch and wave

e phase difference between pitch and heave CFW phase difference between exciting force

and wave

CMW phase difference between exciting moment and wave

t heave at time t ta heave amplitüde

wave elevation at- time t tAB amplitude of absolut bow motion tAs amplitude of absolut stern motion tRB Amplitude of relativ bow motion

X wave length

V w'/g

shallov váter wave number water density

pitch at time t pitch amplitude w circular frequency

we circular frequency of encounter

References

[1] Watanabe, Y.: "On the Theory of Pitch and Heave of a Sh Technology Reports of the Kyushu University, vol. 31, No

1958.

-[2] Gerritsma, J., and Beukelman, W.: "Comparison of Calcula and Measuréd Heaving and Pitching Motions of a Series C5 = 0.7 Ship Mode] in Regular Longitudinal Waves". Labo torium Voor Sheepsbouwkunde Technische Hogeschool D

-Report No. 139, 1966.

-(31 Kim, C. H.: "Hydrodynamic Forces and Moments for Sway and Rolling Cylinders on Water of Finite Depth". Chal University of Technology, Department of Nàval Architect and Marine Engineering, Division of Ship Hydromech.

Report No. 43, April, 1968.

[41 Freakes, W., and Keay, K. L.: "Effects of Shallow Water Ship Motion Parameters in Pitch and Heave". M.I.T. Dep. ment of Naval Athitecture and Marine Engineering, Re.

No. 66-7, Aug. 1966. -

-Ankudinöv, V. K.: "0m störande krafter, som verkar pá

[5]

fartyg vid stampning under rgelbunden söhävning o

grunt farvatten". Leningrads skeppsbyggnadsinstitut, Utg LII, Hydrémekanik och fartygsteori. 1966. (Translation Russian into Swedish.)

Takagi, M.-, and Masaaki, G.: "A Calculation of Finite De

[61

Effect on Ship Motions in Waves". J.Z.K. vol. 122, Dec., Grim, 0., und Kirsch, M.: ,,TR-4-Programm zur Bereehn

[71

der Tauch- ùnd Stampfschwingungen nach der Strei

Methode." Institut für Schiffbau, Hamburg, Jan. 1966. Grim, 0.: Eine Methode für eine genauere Berechnung Tauch- und Stampfbewegungen n glattem Wasser und

We1len.' HSVA-Bericht Nr. 1217, JunI, 1960.

Dickson, A. F.: "Underkeel Clearence". The Journal of

C. H. Kirn

The influence of Water

depth ön the r. idship

bending

moments of a

ship moving in

longitudi-nal regular head

waves

REPRINTED FROM

EUROPEAN SHIPBUILDING

JOURNAL OF THE

SHIP TECHNICAL SOÇIETY

THE INFLUENCE OF WATER DEPTH ON THE MIDSHIP BENDING

MOMENTS OF A SHIP MOVING IN LONGITUDINAL

REGULAR HEAD WAVES

By

C. H. Kim*)

Abstract

The heaving and pitching mOtions and the

mid-ship bending moments of a T-2 tanker model

moving in

longitudinal regular head waves of

shallow water are calculated by Watanabe's strip

theory [1]., [2], [3], [4], [5]. The results are

re-presented in figures and the depth effect is

dis-cussed.

Nomenclature

a, b, e, d, e, g coefficients of heave equation A,B,C,D,E,G coefficients of pitch equation

AW waterplane area

B (x) beam of a section

Cm midship bending moment coefficient

Fa exciting fôrce amplitude

g gravity constant

,Go center of gravity. (C. G.)

h water depth

h wave amplitude

H half-beam draft ratio

i longitudinal rad.

of gyration in%

of L

I W moment of waterplane area

Iyy moment of inertia of the ship about

y-axis

L length betwe perpendiculars

m0

midship bending moment at time t

ma amplitude of midship -bending

moment

Ma exciting mômeñt amplitude

N.

sectional heave damping coefficient

t time.

T draft

T mean draft of a section

V ship velocity

-V displacemeuit 'volume

w

suffix designating wave

X, y, z body oordinates

-x,Y space coordinates

¡3 _{fullness coefficient of a section}

«Europeañ Shipbuilding» No. 1 1969

ECw phase difference between have and

wave. r.

phase difference between pitch and

wave. ,. ...

phase difference between pitch and heave.

a FW phase difference between exciting

force and wave.

CMW phase difference between exciting

moment and wave.

EmW phase difference between midship

bending moment and wave.

C

heave at time t

Ca heave amplitude

Cw . .

waé-.elevation at time t

X wave length

y wave number (o2/g)

vo shallow water wave number

p water density

pitch at time t

1'a pitch amplitudè

(X) circular frequency

circular frequency of encounter Introduction

By applying Watanabe's strip method [1], [2],

[3], [4], [5], the heaving and pitçhing motions as

well as the midship bending moments of a T-2

tanler model moving in regular head waves of

shallow water aÑ calculated'añd the effects of

water depth are discussed.

It is revealed by the calculation that the midship bending moments are inóreásed, while the mothms

heave and pitch are remarkably damped. by the

depth effect.

Definition of ship motions and waves

The coordinate systems here utilized given in

Fig. i are spacer and bodycoordinate system

O-) Chalmers University

of .Technolo, Dparent of

Naval Architecture and Marine Engineering,

«European Shipbuilding» «Noi .i. 1969 z

-(-= vo, tanh v0h

Fig. 1. The space and body, coordinate system.

XYZ and Go-xyz respectively. X-axis lies on the tindisturbed water surfacé and Z-axis pdnts.

vertí-a11y upward, X-axis is longitudinal passing through the center of gravity Go of the ship, while y- and z-axis point port and upward, respectiv1y. The co-ordinate system Go-xyz coincides With the system OXYZ at the initial rest condition. We follow the cònvèhtion of right-handed co6rdinate system.

Asiming only heaving and pitching motions of

a ship at the speed V 'in a longitudinally

approach-ing wavè system, we describe the surface wave as follows

heave añd pitch

A ± B' + C'DE G MacOs(wét"±e'FW)'

4-The coefficients on the left-hand sides of the

above equátions are:-.

a == p V m" dx

b = JN'dx

L

e = 2pgfycbc

L

4 = fm"xdx

L

e = fNxdxVfm"dx

L g

2pgfyxdxVfNdx

L L

A = Iyy+fm"x2d

L

B = fNx2dx'

L

c= 2pgfyx2th--YE',..

-

L..

.,

,

-D= fm"xth

L-= f Nx.dx

± Vf m" dx L L

G = 2pgfyx dx

-L

where p

water density

g gravity constant

V. displacement volume

halfbreadth of a sectión on the calm

watérline ' .

Tyy longitudinal, moment of inertia of the ship's mass àbout Go-y-axis

m" sectional added mass o,f unit

thick-ness for heave

N 'sectional heave 'damping coefficient

of unit thickness

The excitg forces and moments on the

right-bafld sides of the equations (3) are represented in the form

(3)

= h cos (vox + °e t) (1)

where ii- wav amplitude

y0 shallow vÇaterwave number, i. e.

C) '

,» e circular frequenc'of encounter i. 'e.

(+voV)

The heaving and pitching motions of the ship

corresponding to the wave defined above are then expressed by

acos(et+w)

acos(et+aw)

respectively, where are

amplitudes and

w, w

phase differences

be-tween heave and .waye and pitch and wave, respec'

tively.

The couple4 equations and coefficients

The coupled equations of 'heave arid pitch'- of a ship moving in longitudinal regular waves [1],, [2] are written in form:

from [121. If h oò then v

and

Fa

EFW1

-2;J

coshV (h__f{COs vox}

dx

sin e FW J _L ôosh v0h sin vox

wh(w + voV)

m0051T' vo (h_Î5f ces 1ox1 dx

cosh v0h I v0x J

-'

j_f_(» T cosh v0h 1cos v0x

Ma {:.:

}

rif(2rn1_2pgyw) xC

dx

±f(N

dm"

_{coshvo(h_i5JSiflI'oX}

L

_{dx j}

_{cosh v0h} [cos v0x

Jmacos EmWj._ JTacos

_{ywxdx +}

2(

Ima5iflE_mWJ [CaSlrl _CVVJ

l e E

[ac05

_wJ

(f N xdx ± Vf m" dx)] " xdx -+ f.Yxdx)]

+

T°

1J[pgj.

yW x2chV f N xdx - V2 f m" dx

[l'aS1flE -2 (f m" ì2dx + .j:Y x2dx)]

e\.

g

+

TasinWk_fN2dXI

['aC

EçWJ

+ f(_w2m" +

2pgyw)x

cosh vo (h_T)JcOsvoxI _dx

-. cosh v0h J

whf(N_V

)xcoshvo (h 5151flvoxj

_dx.

X _{cosh v0li} v0XJ

5-«European Shiphuilding». Né. i -. .1969 dx

tions of the heaving and pitching motions of a ship

moving in regular head waves of shallow water.

By making use of the rnoiØns. calculated above e

obtain the midship bending moments in the

follow-ing form:

..

.

m0

=

macos (wet + EmW) (4)

where ma and EmW are thé midship beñding

moment amlitud& and

the phase 'difference between wave and bénding. mothent respectively.

Assuming that C. G. lies at the midship sectiOn

the sine and cosine components of the amplitude are written as follows:

where h

water depth

T mean draft of a section

e FW, e MW phase differences bctween

exciting

force. and wave

and exciting moment and

wave, respectively

Sectional values of added mass and damping coef' ficient m" and N for heave are obtained from [6]. In the case of deep water these values are obtained

coshV (h

- T)

cosh v0h are replaced by y and e_ vT, respectively.

The midship bending moments

calcula-«European Shipbuilding» No. 1. 1969

where the integral is taken either hetwee -L/2

and O or 'between +L/2 and O and designates

mass per unit length along the ship length. Dimensionless representation

In representing the calculated results, the follow-ing non-dimensional forms are used:

h/T

depth parameter

À/L wave length to ship length

ratio

V/V

Froude Number

heave amplitude ratio pitch amplitude ratio

Cm ma/pgÌiBL2 Ìnidship bending moment

coefficient where À is a wave léngth. Calculation and discussion

For thé numerical càlculations. we adopt a model of T-2 tanker having the following particulars. Length between perpen

diculars Beam Draft Displacement volume Blodk coefficient Radiús of gyration (L) 3.066m (B) 0.415m (T) 0.183m (Ls) 0.1725m3 (CB) 0.741

(%dL)

0.23

where B(x) beam of' a section

/l(x) fullness, coefficient of a section H(x) hall-beam draft ratio

w/g mass in kg per L/10 (see Fig. 2)

The calculations are carried oút for the following speeds, waves and depths.

-6-140

120 100 60

Fig. 3. Heave amplit!ide ratio at Fn = 0.0.

Station B(x) ß(x) H(x)

w/g

1 0.168 0.442 0.459 6.15 3 0.351 0.749 0.959 12.62 5 0.406 0.911 L115 20.01 7 0.415 0.960 1.184 24.65 .9 0.415 0.980 1.134 22.82 11 0.415 0.980 1.134 24.71 13 0.415 0.980

1.14

24.74 LO 0.397 Ò.961 1.085 21.79 17 0.311 0.871 0.850 13.27 19 O.1Q9 0;837 - 0.298 5.00

F =

À/L =

h/T =

0.0, 0.5, 1.5, , 0.1, 0.6, 1.7, 10.0, 0.2 0.7, 2.0. 4.0, 0.8, 0.9, 1.0, 2.5, and 1.5 1.1, -1.3, LP PP

Fig. 2. Mass distribuUon of T-2 tanker model.

In the calculation the following assumptions are made:

Although trim and parallel sinkage are produced

they are not considered. -

-C. G. lies at midship section.

The Heave and Pitch Amplitudes, together with the phase differences with2espect to the waves are

illustrated in Fig 3-8. In general the motions are

remarkably dampened as the depth decreases. This tendency is more significant as the Froude Number

increases.

l.a Sn l: por 1/10

25

10

0.8 0.6 0.4 0.2 0.0

Fig. 4. Heave amplitude ratio at Fn = 01.

1.0

-7-10,'CT 80

,imgÌ0.

-60

VAS

E r 0.6

.0.4

1111 0.2

ri

60 40 20 o 0.8

«Europeañ Shipbuilding» Nó. 1 - 1969

--10.0,- -06. 08 1.0 1.2 1.4 1.6 1.8 2.0

FÌg. 6, Pitch a.mplitude ratio at Fñ 0.0.

06 08 10 1.2 1.4

Fig. 5. Heave amplitude ratio at Fn 0.2.

0.6 0.8 1 0 - 1.2 1.4 1.6 1.8 2.0

Fig. 7 Pitch amplitude raio at Fn = 0.1.

-100 120 1.2 1.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 O 0.0

«Etnoian Siiipbui1ding» No. ir

-1969

-i-0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

II

-I! -'.5

'

-2.

i

ioo 80 60 40 20 o o -40 -60 -80 00

-8-0 -8-0-8-0 0.02 0 00 1.5 2.0 1.8 I 6 1 4 h 2.5

11

0.01 ,.

L

00 80 60 40 20 -20 -4° -60 -80 100 80 60 40 20 o -20 -40 -60 -80

Fig. 8. Pitch ampThtude ratio at Fn 0.2. Fig. 9. Midship bèding moment at Fn = 0.0.

Q.G 0.8 1.0 1.2 1.4 1.6 I.e 2.0 0 6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Fig. 10. Midship bending moment at Fn 0.1. Fig. 11. Midship bending moment at Fn 0.2.

m'r

h - 1.5

j

0.6 0.8 1.0 1.2 - 1.4 1.6 .1.8 2.0 .8 1.6 1.4 12 10 08 0.6 _1o0 120 -160 -180 1.0 0.8 0.6 .1 2 16 2.0 1.8 14 180 4.? 160 120 100 80 02 0.0 o.oa 0.02 0.01

The Midship Bending Moment8 together with the phase difference with respect- to the waves are

illustrated in Fig. 9-11. The bending.. moments

are generally increased as the depth decreases. This tendency is quite opposite to the above-mentioned

motions. Probably lt is

partly caused by the

decrease of inertial bending moments due to the

dampened motions. The double peaks are nearing each other as thé depth decreìses. This is probably

caused. by the delayed positiOn of the peaks of

heaving mot-ions. Acknowledeinents

The author expresses his thanks to Prf.

Falke-mo, Head of the Division, for his constant support. He wishes to thank Mr. Bennet for his kind advice on the wave bending moments and aLo to Mr. Suiz -for his excellent assistance.

References

[li Wätanabe, Y.: «On the Theory of Pitch and Heive

of a Ship. Technology Reports of the Kyushi Univer-sity, VoI-. 31, No. 1, 1958.

[2] Gerritsm-a, J., & Beukelman, W.: «Comparison of Calculated and Measured Héaving and Pitching

Motions of a Series 60, CB = 0.7 Ship Model in

Regular - Longitudinal Waves.'> Laboration Voor

Sheepsbouwkunde Technische Hogeschool De1ftRe

-port No. 139, 1966.

-[8] Fuku-da, I.: «On the Midship Bending

Moments-a Ship in RegulMoments-ar WMoments-aves.» JournMoments-al of Zosen KiokMoments-ai,

Vol. 110, Dec., 1961.

-[4] Fukuda, I.: «Computer Pro garn Results for Response

Operators of Wave Bending Moment in Regular

Oblique Waves.» Memoirs of the Faculty of Engin-eering, Kuyshi University, Vol. XXVI, No. 2, 1966.

«European Shipbuilding» No. i 1969

[5] Kim, C. H.: «The Influence of Water Depth on the

Heaving and Pitching Motions of a Ship Moving in

Longi.îidiñal Regular Head Waves.'> Division of Ship Hydromechaxucs Report No. 44. Chalmers University

of Technology, June, 1968.

-[61 Kim,, C. H.: «Hydrodynamic Forces and Moments

for Heaving, Swaying and Rolling Cylinders on-Water of Finite Depth> Division of Ship Hydromechamcs Report No. 43. Chalmers- University of Technology,

April-, 1968.

Letweit, M., Mu±Cr, C., Vedeler, B., and Christensen-,

H.: «Wave Loads on a T-2 Tanker Model. The

In-fluence of Variation in Weight Distribution With

Constant Moment of Inertia on Bending Moments in

Regular Waves.» European Shipbuilding, Vol. 10.

1961.

Murdey, D. C.: «On the Double Peaks in Wave

Bend-ing Moment Response Curves.» Adance paper of

R.I.N.A. 1969.

-Joosen, W. P. A., and Waliab, R.: «Vertical Motions and Bending Moments in Regular Waves.- A Corn-pari.son Between Calculation and Experirrnt.> I.S.P. VoL 15, Jan.-, 1968.

[101 Ivarsson, A., and Thomson, O.: «Jämförelse mellan Modellförsök och Beräknade Värdeñ för Fart ygs Upp-- trädande i Regelbunden Vägor.» Chalmers Tekniska

Högskola, Institutionen för Skeppsbyggnadsteknikk,

Sept., 1965. -

-[il] Grim, O., und Kirsch, M.: «TR-4 Programm zur Be-rechnung der Tauch- und Stampfschwingungen nach der Streifen-Methode.» Institut für Schiffbim, Ham-burg, Jan.-, 1966.

Grim, O.: «Eine Methode für eine genauere Berech-nung der Tauch- und Stamp fbewegungen in glattem Wasser und in Wellen.-» HSVA-Bericht Nr. 1217, June, 1960.

Dickson, A. F.: «Underkeel Clearence.> The Journal

of the Institute - of Navigatdn, Vol. 20, No. 4, Oct.