On the theory of pitch and heave of a ship by Y. Watanabe. Translation

(1)

ON _{HE THEORY OF PITCH AND HEAVE OF}

_{A SHIP.}

By Y. Watanabe,

Engelse vertaljng T, Sonada I

maart 1963.

Rapport No, 1O,

Teehnolog7 Peorts of the Kyuhu

_University.

Vol. ,31 No

1.

janua2 i95.

1, Iflt!Odutiofl,

Xorvjn-KyouJcove

_{has atudled the heave and pitch motion of a}

8hip in waves alying Munk'

_{method. which was used by}

_{him to}

calculate the force applied to

_{an air-.ehip. The author is very}

much interested in this

_{method, but it is a pity that}

_{Nunk's theory}

has not been explained clear

_{enough. Therefore the author}

_{tried to}

retreat his method and to make

_{it ease to use in this}

_papere

2. Free heave and free pitch.

Zeverai. methods has been

_{studied to calculate the force}

_applied

to a ship by solving the

_{equation of the motion of fluJ.d along the}

bull in heave and pitch. These

_{methods are very complicated and the}

author believes that the strip

_{method is more convenient for}

practi-cal purposes.

(2)

OX is a fixed axis ori the still water surface and OZ is the

ver-tical axis on the surface in Fig. 1. G, the centre of gravity is seau

med to be on the water surface without losing generality. Ox denotes the axis f xed to the boat, positive foraward of and 07. denotes the axis positive downward. XZ agis coinoidse with XZ at t O. The boat

han the velocityV into the direction of +X. Let be the pitch angle and

the heave

displacement The position of the boat

at t

t is

euch as described in Fig. 1.

Suppose is small:

XVt+x

(1)

Assuming AB. as the controle plane fixed in space and the distance

dX . dx. The force applied to the boat in the above mentioned condi-tion will be as follows r

(A). Neglecting the effect of free water surface.

a, The force associated from the relative motion of a ship and water

b. The force from the statical

water pressure.

(e).

The force associated from the effect of free water surface. The author bas tried to apply these forces to the strip dx in the direction of X. The change of motion of the water in X-direction along 4C is small provided that the boat is very slender and proceeds with the velocity ', while change of motion of water in the

trans-verse

direction at dX is

large. Therefore, the change of motion

in

the direction of X axis is neglected and only

the motion on the

ver-tical surface to X axis is studied.

(A). Neglecting the effect of free water surface.

a. The force associated from the relative motion.

(aJ

3

(3)

-3-,

The strip has the downward velocity of

xt.

The section ₁ at AB at t = t is replaced by ₂ after , t because of pitch angle

Therefore section 01 aoves downward by q dt at AB during the time

4

t and it has the downward velocity cf

_{V Ø.}

Total downward

velo-city V5 is expressed as follows:

q

V8=

-x

+Vø

(2)

in the mean time the sectional form of AB changes. Suppose the body described in fig. (b) moves with the velocity of V8 nd neglecting

the effect of waves on the water surface, the water has the momentuit

S1

_V

in the direction of

V*

1 18 expressed

x

and

f' S

is called "added sass".

(&)

This momentum coincides with the one under

the influence of free

water

surface

for high frequencies. Thus the force corresponding to the change of momentum is applied to the boat in the direction of

Let this torce dF55/dX:

dF55

s1vß]

+ v

Assuming the ship to be

wall-sided,

X =

¿onetant at AB.

Then

formu-la

(i) is as follows:

dt

4t

dt

and:

-x 2V

(4)

It is not correct to neglect the term d81/dt as done by

J.A. Fay (Journal of Ship Research, March 1958), b. The force of Btaticd water pressure.

At £8 the body is in the downward position by ( -xe) from its

equIlibrium position.

Assuming the ship side to be wall sided and let b the breadth of the ship at AB, the downward force expressed as follows is

applied: dF

(k)

w is weight per unit length.

(8). The force associated from the effect of free water surface.

Equation (3) and () are derived by neglecting the influence of free water surface. There are two influences resulted from waves on water aurface

In

the first place, added mase changes,

_The correction for

this change is done by multiplying S

.

by cofficient C2, which is the function of (bW2/2 g). Thus:

82 C2S1 = Q1G280

In the second place, there is the resistance against the motion, namely, the damping of the ships energy by propagation of waves. There are frictional and eddy making resistance besides above mentioned resistance. And it is a well known fact that the wave ma-king resistanc. dominates in the total resistance in case of heave and pitch motions. Therefore, only the wave asking resistance is

treated in this paper.

R.ferring to

Holstein, the wave

making resistance is expres-sed against velocity V5 as follows:

22

-2

(5)

where:

&circu1ar frequency of the motion.

(5)

and:

,

12 _z

-dx =

xS2

-5

Integrating (6) for the ships length1 the total. force originating

heave motion F5 is expreosed as follows:

F8

_-

pf

s2 ax +

fcs,

r

_[

.

f

.fx4x1

f Xax}

dx]

*A5 +gm

(7)

where:

water plane area

eoment of water plana area,

and:

j

dS2

_(bow)

_(stern)

j dza

let the sectional area

O it both ends, then:

ras

J -dx= o,

L

e

ratio between tua amplitudes of waves originated and

that of motion and expreaed as a function of

b2/2g.

This function depende on sectional forms and Grir has tried

to give come of these foraula.

Summing up (3), (k) arid (5), the force applied to th unit

-length is exreaaod as followe:

=

°

[s G2

xÇS2 +

+

x$E

+ gb

- gbx

+ (g8 -

(6)

-6-M denotea the total added aBB over the ship and is derived.by

making the correction for the tree water surface

2 to the ordinari

vibration terms The value for this correction is '' 0.75.

=

.4Mg

_+ID

_(Ix

5a

dx)

[AM V-.

fXj2

dx)

[I

dx..Ç'VZ

L

dx*-og

+g

(8)

The aoment -M8 whïcb originates pitch is obtained

by iiultiplying

b3r (6) and integrating over the length.

=

fS2dx

+

+eÇ1X2z2dX,

2 _{dx +}

_VM]

_{_v4vM+}

e-dx]

_ogm5+giwt _gV0 Ç

(9)

and:

_dZ

f2

dx = -2

fx

S2 dx,

1.

where:

iwV ¡

0*

= displacenient in still water

0

O

Iadded maSa moment of inertia including the effect of tree

water surfaces

iwmoment of inertia of water plane area

Let M0

= mae

of the ship, I, = moment of inertia of ship and eq.

of free heave arid pitch are as follows:

NF

o

s

=M

Na.me].y, sg4 of free heave is:

(M [

f

,c2dx

-MI?

4 j

¡24

-

4g

_$

dx 3

=0

(10)

-7--i

(7)

Let GM

_bf!e

eq,, of free pitch is as follows;

0»

[1x32 dx]

+i!Ç 1X212

_dxl

..[eÇ

=0

Assuming the ship to be symmetrical against G, (lo) and (li) are

ex-pressed as follows;

heave: (M +4M)

+

f

L

2 +4MV

(D L

¡2

_{dxi, +}

A5

¡2

_{= o}

pitch: (I

i-AI)

+

dx

₊

[rBMe

_iv2]sv1AM

₌₀

+ à MV

in eq

(12) and - AMV. in eq. (13) are terms for dynamical

coupling.

There are terms in eq

(12) and (13) depending on V. Usually

ve-locity is treated as

V =

O.

If Y

O there is sorno effect in the

eq. because of the above mentioned terms,

3

Forced heave and pitch motion on waves.

The .q. of motion in question iB treated as well as the free

motion instill water making the correction due to wave action in

(A) a.b. and (B). That is the relatiVe velocity between waves and

the ship IÇ

V + V

instead of V5 is introduced in the momentum of

(A).a. (vw denotes the upward velocity of wave). The correction due

to wave action is made to the statical buoyancy in (A)b.

instead

of V is introduced neq, (5) in (B).

The characteristics of wave ought to be made clear. Put the top

of waves to X = O at t * O and let the Wave height be b. The

(8)

(c)

+1 with velocity C is given as follows:

h oós

(X

Ct).

8ubetituting (1):

h cou .- {x

_-

- V)

tJ

or putting

= K, 0

21 (C

V)

'- (T

±s period of encounter),

h coo (lCx -set).

The eq. of motion of water p&ricle at the depth

in still water

is as follows:

b e

coe(KX - 4)t)

con(Kx

û)e t)

where

I

27t

z

The upward velocity v of water particle due to wave action io

gi-ven by different&ating

_'2

with t. (X = conet.):

27

=

= h0e

ain(Kx-.

)et)

(15)

Although X 'z oonat. v

changes slowly according to

.

o Therefore

io asounied constant

.tàkiiig

the mean value. The mean draft for

¡î is introduced

as the mean value of

. Therefore

=

Then in the part of (A) a, (B) the following Vis introduced

ins-stead of

(9)

or: d dx dx

C2

-w2i_gbhe"mcos(Kx_Gi)et)

dx

27t

=

(g2

=.- h WeX

m

e

B2

«e3

oin(Kx-4)et),,

27 dF

e

coa(Xx-)et)

₂

2i

-+ çhWVe

X

sinKx-

Wet)

_dx

2 the total force due to wave action te obtained by summing up,

(where

m

e),

e [(gb_r2s2) [cosKxco

Wet

-= -be

2-2

dS

jA

+sin(Kz)sin&et}

_[3

- dx

ÇsinKxcoa4et

-9-The $roude-Kriloff Hypothesis is applied for (A)

b. The water

pressure at the bottom at = O is approximated to the water pressure at the pert of which dtaft is

m = 30/b.

And this is approximated to the increase of water pressure as much as the water head of wave height,

neglecting the influence of circular motion at water particle. There fore the increase of bwyancy due to wave action is expressed as follows, (shii side to be wall sided):

2i

gb

e

m

,Dgbb e

oos(Xz- &et)

(17)

The forces for unit length in ec (3) and (5) are corrected due to the second term in eq0 (16) and (17) for eq. (k). This correction denotes the

torce dP/dx due to wave action:

dx

(10)

--. cos K x am

'. t

The force

orig3.nating heave motion is obtained by integrationi

= ...,oh

_[[g

[b._ecoexxdx_iP2 1820_eCOØKXdX

e9eiuKxdx

+

fbe68inKx

dx-1#2

sin Kzdx

_4'

fr_

ooaKzdx +WV

j

We L L

_J

ABauming the ship to be 8ymmotric«, the third term for coe:4Mt

asid the firet, second and forth term for ein &e t are swept out.

_h[Çgfbeco8Kxdx_W2fS2

_e

_coexdx

-

wV

fe_e

-

sin I(x dx

J

coe&et

L

¿f2S..eOKXdXBiflW.t}

(19)

¿de3

L 27CC

_e

and let 8

em

°

.d (d is draft) for e

=_b._9m[{g fbcoaKxdx_w.I_Rv)

L

¿.fr2

41e3 L

where:ç

J

ainKrdx=K

L

_ lo

-cos Kzdx sin

4et

1 S2 ces Kx dz

2cos

Kxdxjcoo44t

(20)

The second term for cos 47e t in (20) ie

eeu1ted from v

in

(A) a, the first term from (L) b

and the terms for sin é7et is z'e

(11)

The moment 14, originating pitch motiona is obtained by multi-plying x to (18) and by integration. Let Q Q and the shl.pform to

be symmetrical:

+M=p[f

f

oeKx2J2osKxdx

e0ainKxdx

'v

Ix

esinKxdxoo8e4e t

L

+gfxbe6jnKx4x!..Ø fx2eainKxdx

_(L

L

2f

f

d(&)

i

4eL

L J

MW=,be[&..3x2einKxdxco8Wet.tgfxbeinKxdx

-(9V

¡82eoaKxdx-I(*/EY)

fzs2si

Xxdx}ein Wet]

The terme tar ooe ¿sse t

_{re resulted from (B) and the first term}

for am

&Je t troni (A) b, and the second and

_{third terms from (A) a.}

According to Gerritama, the terms resulted from (B)

_{can be neglected}

but the terms from

CAJa. cannot be neglected and the terms from (A)b

are of most importance.

fb

cos Kx dx

.W(W-K) f2

coslxdxJcoe&et

M=

x

atri Rx dx

1ain

/'e t

J

- 12

this

eq. is expressed as folløwa:

p'

f

(12)

12

-Therefore the eq of heave and pitch motion on waves are ob-tained by rewriting W to &'e in eq, (12) and (13) and. substituting (22) to the right hand aide of these eq,

The longitudinal wave bending moment ia obtained by integra-tirg eq (21) and (22) for the half length of the ship. Effective wave

length he

he

is

introduced instead of h in (22).

According tot Hariaoka V has little influence to those motions,

these eq,

of motion will, then,

be more simplified. Putting t = O and neglecting the influence of water particle (terms for

these eq, coincide with those derived by Gerritema (I.S11',,1956), SisaSç Williams (T.I.N.A., 1956) and Weinbium 1950).