The influence of ship speed on pitching, heaving, and hull deflection in waves

B.S.R.A. Tran&oi No 284

THE UPL1EiCE OP SHIP SPEED CN PITCHING, HEAVING, AND HULL DEPLECTION IN WLThS

D.M. Ros tovteov

Trane. Leningrad Shipbuild. Inst., No.46 (1964), p 77.

(R.port d.liv.r.d at the Soi.ntifio Conferenos of th. Inetitute held in Deo.mber, 1962)

Let us consider the pitohing and heaving of a ship moving at a oonstant speed directly into the waves, with relation to a system of plan., regular waves. The 000rdinatee z, y and z are related to the ship (Pig, i). The fixed system of coordinates

and4

coinoides at the Initial moment of time with the system z, y and z. We will denote up end down movement of the ship in terms of , and ita differenoe, the plus direction of whioh is shown in Fig. i , in terme of _f' The equations for surging by a ship moving at a constant speed in a regular wave pattern, compiled using A.N.Krylov'e theory and allowing for

the

effeots of ad3oining maeses by P.F.Papkovioh'$ method, take th. following forms

(it + it1) + )

+1F

_{+ +}

aq''

S '. P oo( o't + ), (1)

I.

here M and M. are the mass of the ship and the oonneoted mass of water;

K and K1 are the moments of inertia of these masses;

AÇ and are the ooeffioientø of damping of th. heaving and pitching;

almost the same as one another, we reach the oonolu.sion that the

interconnection between heaving and pitching, determined using equations (i), je extremely indefinite. Experimental research has aleo shown that there is a definite relationship between the vertical movements by a ship and its pitohing movements. As an example, the results of measuring the movement amplitudes using a model are given in Fig. 2, also calculated data based on using equations (i) separately and together; all the ooeffioients for the equations were determined by experimenting, with different towing conditions. It follows from Fig. 2 that heaving depends greatly on pitohing, and this oannot be explained on the basis of equations (i),

This article contains the necessary ohanges in the pitching and heaving equations, based on the more accurate determination of the foroes aoting on a ship moving forwards; the effeots of these changes on the heaving and pitohing amplitudes, also on the bending moment at the midship section, are estimated.

In reference (2, M.D.Khaskind considered the heaving and pitching problem for a ship, using hydrodynamic methods; he proved that, with a ship moving at a forward speed u - oonet., an additional vertiosi hydrodynamio force acte on it (by oomparison with the oase in which

o) Ò(Q

1d9

2

1dSJ

ue

iat

, (2) - [

èx

òn

s

èx

n whéra e

is the mase density of the water;

v, are the amplitudes of the vertical and angular speeds of the ship during heaving and pitching;

S is the wetted area of the ship;

and

2 are the potentials of the induoed speeds when the surfaoe S moves with identioal speeds in the directions corresponding to heaving and pitohing;

We know that allowing for the foroe (2) resuited in a substantial ohange in the conditions for the oonneotion between pitching and

heaving. In partioular, the equations proved to be linked for a ship Bymmetrical relative to ita midship seotion [2].

Later, when he was working out praotioal methods of calculating :the parameters of heaving and pitching on the basis of solving the problem for a cylindrical ship, in referenoe

[3]

M.D.Khaskind permitted inaoouraoy to creep in when he assumed that the potential of the

induced speeds, when heaving and pitohing were made to 000ur in still water, do not depend on z, The result of this was that the type (2)

term was missing from his equation for the hydrodynamio forces. For a cylindrical ship we oan write*

i

-i(');

2

dS- dx

dl,

(3)

where dl is an element of the outline of frame station

On the basis of equatiie (3), we get the following expression from equation (2) for the intensity of the additional running badi

iot

- - - -

_{I e}

dl.

dx

Por pitohing we have i

jort

Ue

_{_*1.}

Following the reasoning of M.D.Xhaakind [3], we further writei

w1 i

dl -

33(X)

where

(.&33(x) and

.X33(x) are the running oonneoted mass and coef-ficient of damping.

On the basta of equations

₍₅₎

and (6), we get the following from equation (4)s

Aq(x)

_{- uJ[t433(x)}

(7)

(6)

(4)

Noting also that

i°-t

instead of equ.ation

(7)

we gets

¿q(x)

- u

33(X) + u

' )33(x).

₍₈₎

Let us clarify the physical sense of expression (8) for the intensity of the additiaial hydrodynamio load.

Let US oonsider the seotion at a distance z from the midship section, with a differenoe O (Fig.

3).

We have assumed that the ship has a horizontal speed u - oonst., whioh resulte in the additional transverse seotion speed (in the plane of this section)

- u,

(9)

and the additional acceleration relative to the surface of the water

Air

- u(,.

(io)

If we compare equations

₍₉₎

and

(io)

with equation (8), we find that equation (8), and also of course equation (2), allow for the effect of the additional speed and aooeleration of the transverse section of the ship relative to the surface of the water, this speed and acoeleration developing while the ship is moving forwards, on the hydrodynamio

forces.

In order to estimate the effect exerted by the additional running load (8) on bending the hull of the ship when waves are encountered, let us ooneider the heaving and pitohing of the symmetrical midship plane of a ship.

The total intensity of the load in the section z

i8

determined by means of the equation

q(z)

- [m(x)+

'33(X)](

+ x) - 33(X)(+ x4) +

+)'33()

+Ïr0b(x2(k0T)[1

b(x)

where, in addition to the symbols uaed.before, we introduce the followings

b(x) is the width of the waterline at seotion z;

(ii)

is a oorreotion for the amoothing of wave movements a8 the depth increases;

k0 - 2 -ir/), the wave number.

If we write the heaving and pitching equation in the form

0.5 L

.1 q(x)dx 0,

-0.5 L

and take into a000unt the symmetry of the ahi;, we gets

where

o

+

À4Ç+'Fç_M1v,...À

vif'.

2r0X.2(k0T)

f

b(x) [i -2 n2 K + K1

0.5 L

f

q(x)xdx - 0,

-05 L

2 Y2 (K + K1

)2

-0.5 L

o

A

f

- -

2r0

f

_xb(x)[1- O Jam k0xdx ein ct.

-0.5L

Tb(x)

It followa foin equation (13) that the equations for the heaving and pitohing of a symmetrical ship prove to be linked when the

additional foroes depending on the forward speed are taken into a000unt. There is a relationship between heaving and _{pitohirig, and}

for a symmetrical ship there la no reoiprooal oonneotion, If we write the second of equations (13) in _{the form}

'rr0BL2 Oietm t, 1 ₍₁₄₎

O b()

L33OØ L

_2x

-

x2(kT)

f

[1

-1 B

b()

J8inTir.d, and

L

we find ita 80]UtjOfl

4).

(q)1 ₊

iV2)e.

₍₁₅₎

(12)

/433

008

k0xd.x oos O t;

'b(x)

dynamio ooeffioiente.

On the strength of equations (15) and (16), the first of the equations (13) oan be transformed as followas

+ 2'n

r0BL

[8

y(R1 + iR2)Jett

M + M1 where while where

We oan quite easily write the following equati2s

2 nl -I M + M1 n4L(M + M1 R1 -2

rl

o3

i«i

y

1 BL2 -

_{r0 ;}

1 2 y

2Y2d2

1

here -

,and

2

22

(1-d)2+4rd

(l_)2+4r2d2

i

n1L(M+M1)

R2

-e

(21d2+d)

y

°

_b(s)

V33o

_j

OOBTTkÇd

2(k0T) J

[1

B

)b()

-1

The solution to equation (ii) oan be found in the form

i o-t

Ç. Çe

0l 02'

-r0---&(1 +ip2)

, ('7) (19)

(20)

-, (M + M1 )n (16) are the

and ci. is the waterline ooeffioient,

The effects of the additional foroes considered above on the amplitude of heaving can be estimated from the magnitude of the amplitude ratio

02

si

Ç 01

On the basis of equations (16)(22), we get the following equation

for a syetrioal ships

02

i

6

L(M + M1)1

t/d

+ 4\f

(23) k

- -

T

(i +

/(id)2

+ 4d

We can see from equation

₍₂₃₎

that as the pitohing comes closer to resonance (d2 . i), the ratio k mareases,

To obtain a quantitative estimate, the ratio k was calculated for a symmetrical model whioh had been tested for defleotion in the tank at the Leningrad Shipbuilding Institute (LSI). The dimensions of the model weres L - 2.08 m; B - 0.32 m; T - 0.112 m; o( - 0.67;

- 49.8 cg.

Two oases of loading the model were considered s still water bending M - - DL;

8W

still water flexing M8

-Table 1 oontaina the resulta of caloulating the ratio

₍₂₃₎

for f our towing speeds.

P2

-(i -

d)2

+ 4')d

(22) ia the oornplex amplitude of heaving and pitching without allowing for the additional forces aasooiated with the 8hip'8 forward speed;

02 - r0

k-u

[R1 e2 + R2 + i(R2 2

+ R1 8)]

(21)

is the additional complex amplitude of heaving, proportional to the speed u.

In equations (20) and (21) we have used the following symboles

2

-ti, r2/seo 0.4 0.8 1.2 1.6

u

Fr - 0.089 0,178 0.266 0,356

Bending in still water, k 0.412 1.45 2.15 1,38

Flexing in still water, k _0.494 1.33 1.39 _1.23

t-r

_Obd

---o1.

[oo

where

1

y

TABLE i

It follows from Table i that, if allowance is not made for the additional force8 oonsidered earlier, the error in the magnitude of heaving beomes extremely great.

Ir we oonvert to real quantities in equations (16) and (22), we obtain the solution, as regards the keaving and pitohing of a symmetrioal 8hip, in the following forme

r0 BL3

'y---

sin (o-t_i, ),

(24)

L2

(c't

) -k cos (t

(25)

+ 4d

ç -

(1_d)2

₊

_42d2

2)d

2 tan

tan4

- i - (26) tan 8

-d1 sin + 2'Yi coB

d1

008

_{2'i sin}

In order to estimate the effects of the additional forces associated with the speed on the wwve bending moment, we calculate this for the midship section of a symmetrioal ship. The midship section bending moment is equal to

-0.5 L 11 - , ₁₂ -M o 2

f

33(x)xdx M1 M2(0) - 1r0BL2 1 n1L

[(A2B2-A1B1)oos at + (A2B1+A1B2) sin otJ, (31)

o

M(0) .

_{- ¡}

q(z) xdx. (27)

0.5 L

If we substitute equation (12) in equation (27), and use equation (13) for the pitching and heaving of a Byumetrioa1 ship, we gets

M(o) - _{(11_12)} (i-2

₀₀₈

_{Bin ot)+}

(1_l2)[(

2

uc2) 008 at (cr2t,2 ui-'1) ein otJ

-2

2u 'i°°

O't o - ein otJ + r0

008 Ot ¡

%2(k0T)[)b(x)

--0.5 L

-

A

'3](j

-

_{12)006 k0 xdx, (28)}

where in addition to the symbols already employed, we have also u8ed

o

o

2

f

rn(x)xdx 2

f

'b(x)xdx -0,5 L F (29) 2

f

À33(x)xdx -0.5 L .

Using equations (16), (20) and (21), we find

M(0) - M1(0) + ₍₃₀₎

where

M1(o)

is the midship section bending moment, calculated not

allowing for the effects of the additional forces associated with the ship's speed;

M2(0) is the bending moment proportional to the forward speed of the ship.

The moment M1(0) can be oaloulated using the equation given in reference [4],

where M

_\'21

i

Al_M

L

_1+?'

lt

₂ 12 2 Yi L i

d(1d)(2

Y1d(2 Y1

1d1) -1+

(1_)2

_{+ 4} M

d(ld)(2

_{'1' ci2+a(141)+2?1d(2d1d1...} 2d1) B1 B2 -2 where M2(o) -r0BL2

The morflent M1(0) can be written in a similar form.

Table 2 contains values of the dimensionless amplitudes of the bending moments M1(0) and

M(0)

, also the phase angles

'2

oaloulated for the model the dimensions of whioh were given earlier. , the dimensionless bending moment,

TABLE 2 u, rn/eec _{0.4 0.8 1.2 1.6} u Fr

0.089

_0.178

0.266 _0.356 0.0448 0.0391 0.0402 IM (0)1 2 0.00272 0.00458 0.0020 0,0013 ¡M2(0)g: I 0,06 0.12 0.05

23

28 _{63 14} 0.0435 0.0448 0.047

o

0.0035 0,0033 0,0012 0,0010 M2(o)1 _0.08

_0,07

_0.02

3.0

79

79 19 M M + 4' + d

Equation (31) oan be reduced to _{a form oonvenient for ita uet}

M2(0) - M2(0) j

It follows from Table 2 that, whatever the pitching and heaving oonditions, the amplitude of the ad&itiona]. bending moment M2(0) _is 'small by comparison with that of the moment M1(0), The amplitude of

the moment M2(0) depends very largely on the magriitue of A1, calculated Bing equation (32). _{In the oase under oonsideration the parameter A1}

liad the following values s

i A1 - - 0.0337 with M - - - DL; Bw 60 i A1 . 0.0021 with

M

- - - DL.

60

The particular example oonsidered does not enable exhaustive conolusions to be drawn regarding the quantitative effect of the additional foroes associated with the forward speed of a ship on the bending moment, _{One would, however, expeot this effect to be appreciable} in the case of a ship with a bend in still water.

M i

If the flexing moment in still water _{( -}

_{> - )}

i Substantial, DL 60

the effects of the additional forces mentioned above _{i the wave bending} moment in the midship seotion will always be infinitesimal _{(of the}

oìder of 1,0-3,0% of ¡ M1(o) j ).

On the whole, the solution oonsidered above has shown _{that. heaving,}

and under certain conditions the wave bending moment in the midship section, are greatly related to the foroes proportional to the forward speed of the ship, and these forces cannot be ignored when the conversion functions for heaving, pitching and the bending mouient are being determined.

It must, at the same time, be noted that the solution obtained above i8 only effeotive for a ship moving at a constant speed. _{In actual fact,} wlien a ship is passing through waves, its speed alters periodically owing to the effects of pitohirig and heaving, the wave train, and the wind. The quantitative estimate8 made above may be appreoiably altered owing to

this. _{The plan for further developing the problems considered in this}

article includes the extremely important task of solving the problem of

pitching and heaving for a ship moving with a constant restraint, this

In conclusion, let us write equations for the heavingand pitohing of a ship asmmetrioal relative to its midship seotion.

After substituting equation (ii) in equaticzis (12), we gets

(M+M1) + + F +S +[À

_{-uJ +(}

F- ?)

_F0ooe(t+S

) (34

+(À_usM+M

_)+(y

u)+sM+M +

Àe+YSF

-0oos(at+

The oompari3on of equations

(34)

and (i) shOws that allowing for the ad.itional foroec associated with the Bhip's forward speed only greatly alters the extent to whioli heaving is related to pitching. The

terms in the eeod of equations

₍₃₄₎

dèfining the effects of heaving on pitching ooinoide precisely with the corresponding terms in equations (i), Consequently the relationship of pitching to heaving remains extremely indefinite. By comparison with equations (i), oertain of the terms in the equation for pitohing are slightly different in equations (34), From the quantitative point of view, howvrer, these changes are extreraely email (not more that 10% of the corresponding basio term), The prinoipal

features of the combined effects of the angular and vertioal movements

of ships during heaving and pitching, whioh oan be found by analysing 'i

equations (34), ooinoide preoiaely with the results of the experiments mentioned earlier (Fig. 2).

Then the order of the different terms in equations

₍₃₄₎

is assessed, i!t is found that the surging of an asymmetrical ship oan be calôulated, with an error not exoeeding 10%, using equations (13), on oondi-tion that

he right hand sides of these equations are replaced by the following

expressions z

in the equation for heaving,

0.5 L

_{\-3(x) cr}

r0X.2(ic0T)

_¡

_b(x)[1 J oos(Ct + k0x)d.x

;

(35)

-14

in the equation for pitohing,

0.5 L

A33(x)Yg

r0-2(k0T) ¡

b(x)[1

_-

_J

008

-.0.5 L

_b(x)

REFE REN CES

GERRITSMA.

Ship motions in longitudinal

_waves.

_{International}

Shipbuilding Progress, 1960, 66.

M.D.IcR&S}CIND,

_{The hydrodrnamio th2ory for the pitohing}

_and

heaving of ships in waves.

_{PMM, vol,X, ist edition, 1946.}

M.D.KEASKIND.

_{Approximate me-thods of determining the}

_hydrodynamic

particulars of heaving and pitohing.

_{Izv. AN SSSR, OTN, 1954, No.11.}

L.M.ROSTOVTSEV.

_{Determining wave beñding urnents.}

_Trudy

LSI, 1960, XXXII.

ut

t-n- iiC

2AIj

i Oc4WtWMfl Ae4A.2L ffo.c

ik

Fig. 2.

t..

L

-

15

z

pacvem naCOO*J#NJPf YPOWWUR #CCt%i1 CVt,,? 1O jÒVHW WURM gQVKU

---1 \v

/xi

-

experiment

- oaloulation by combined equations for heaving and pitohing.

- oaloulation by the equations for heaving: and pitohing separately.

-