The steering of ships in following seas

(1)

ARC'

I-I

A

ME

4,

'!NG

I

Experirnentol Towing Tonk

Stevens Institute of Technology

Hoboken, New Jersey

Lb. v Schep

Technsch

Technical Memor'

No.94

April 1

THE STEERING OF SHIPS

IN FOLLOWING SEAS

PREPARED FOR PRESENTATION AT THE

INTERNATIONAL CONGRESS OF APPLIED MECHANICS

LONDON, SEPTEMBER 1948

by

(2)

April 19S0

Experimental Towing Ta,k

Stevens Institut of TeçhnOlor

Hoboken, New. Jersey

TI Sfl OF SHIPS

IN FOLLOWING SEAS

Prepared for Presentation at the

VII International Congress of Applied Mechanics London, September i918

by

Kenneth S.L Davidson

(3)

TABLE OF CO

PAGE

ABSTRACT

I. . INTRODUCTION

II.

PRLDINA

CONSIDtATIONS

III. FORCES AND }4CENTS . .

Forces and Moments Imposed by FollOwing Seas

o3ymic P'opies of the .5hip

.

1 2

3. 3 8

Rudder Force and Moment . 10

IV.

STATIC EUflRIUM

. 1].

V. DYNAMIC CONSIDATIONS . 12

uations of Motion .

-trnandc Stability ozi Course inStill Water;

12

Rüddér Aidships -. ₁₃

Motion in the Following Seas; Rudder Fixed 114

VI. CONCLUDING REMARKS . .

16

19

LIST OF CAPTIONS . - 20

(4)

ABSTRACT

Previous studies of the steering of- ships have dealt, for the most

part, with still water. These should be .regaded as. the initial phase of a more- complete treatment of the. subject. Their purpose-is to.

evalu-ate steering quSlities in .re1ation'to'designa Theôamiot be-presumed

adequate to . determine desired steering qualities, or. the limitations of

particular steering qu4ties, in terms of

performance.-The steering qualities of seagoing ships need .to be .emined .also.

for the effects of rough'water, which may impose more séere requirements

'still. water.

The present. paper discusses in- preliminary fashion an important..

aspect of steering in rough water; namely, steering in large following seas where broaching-to is a possibilit7o - The.' simplifying assumptions 'are made that the seas are of simple 'trochoidaj. form .:with wave lengths

comparable -.to.the length of the: ship, and that the., ship. moves ixiitiaJJy

at thewave Epeed given by the usuaJ. theory. . Consideration is given to..

the.abiity of the rudderto prevent changes of heading, and to the motion subsequent .io-.a.diturbance with the.ider angl-e.fixed.

The model of a particular destroyer is used as an example,. with waves equal in length to the model length and of 1/20 height-'length..rat.io

it is shown that thern full rtidder angle of

_35°

is needed to prevent changes of heading--when the angle- between the course and. a normal to the

wave-crests-reaches. 209, and that. at smaller angleS an-accidental- small-

dis-turbance causes- the. heading todiverge. rapidly, with fixed rttdder angle.,

(5)

I.'. INTRODUCTION

It is well known. from practical experience that following or-quarbering seas -iiose -a. criticaltest. of steering qualities..- Most

-ships are harder tosteer-.when the seas are behind-them thanunder

aIv-other circumstances.. In extreme conditions., involving a high ship speed

in combination with really heavy following seas, it Is- possible for

steering control to be lost coletely so that the ship slews around

broadside-on and. broaches-to. _{This sometimes happened to sailing vessels}

running before a rising storm, the dndáge of even their bare poles and rigging preventing their being held down to safe speeds.

Previous studies of steering have dealt, for the most part, with

still water. In a recent paper experimental data for six representa-.

tive ship types obtained from model tests in still water were applied to the equations of motion to deduce values of the

dynamic

stability on

course- with rudders amidships . The equations of motion were derived

in

the form reproduced--here as uations (9);. their solutIon is given by uations .(10). The two exponents. p1 and p2. were found to be real for-. all .. six ships, with much smaller than p2. The va]ue of p1 was adopted as the index of stability, and is shown in Figure 1 for the six ships.

This chart, which is taken from reference 1.; shows also the. unfavorable

effect of increased stability on the minimum turning diameter with

hard-over helm.

It was concluded in reference 1 that 'on analytical grounds steering

ought to be improved by increased dynamic stability, and that full-scale

experience with the. six ships generally confirmed this. It was pointed out at the same. time, however, that the least stable (actially- unstable)

of the six ships, the minesweeper (A), could

in

fact be successfully steered by hand, at least in ordinary weather, - Thus, although the range

of the stability index 'covered in conventional designing had apparently been established reasonably well, the true significance of this range 'was-unknown, and evidence was lacking regarding the values 'of the index really

needed -- or highly desirable -

in

practice. Evidently, to. gain further-insight, additional factors would, need to be taken into account0 The most obvious first choice was the influence, of following .

TM-9I

(6)

-1-TM-91

The work described here is: purely exploratory in character, and was

undertaken as a first ste,. mainly in an atteit..tO. gain an over-aJ.1 viöw

of probable orders of magnitude for the drnamic effects of following seas

on the motion with rudder fixed, and on the effectiveness of the rudder

for control0

To be definite, the destroyer (F) of Figure 1 is taken as an

exai1e.- The ship is assumed to. be traveling at wavespeed in following

(trochoidai) seas: of her ow-n length and 1/20. height-length rabio.;: the

prob-len is .thas reduced to one of essenti]Iy steady conditions0

Heeling of

the ship is disregarded, as having .only. .a .secondary effect. on the forces

and moments governing steering tinder the assumed conditions0

Ii.

I

CONSItRATI0N

The photographs in Figures 2 and 3 show a mOdel of.. the ship

..

during

crude tests on straight course to determine . the inf1uence of following

seas on the latera' force

turning moment produced by yawing.0

.

The

tests embraced 1/20 waves having lengths from

3fJ

to .1-1/2 . tines the mOdel

length, the model speed being adjusted as necessary to match the wave

speed0

The model was self-propelled, but attached to the..towing carriage

of the tank to provide for restraint in yaw and for measurement of the

force and moment0

The carriage irtroduced longiturinA) restraint :as well,

and since precise regulation of its speed to match the wave speed was

some-what difficult, a gradua.l shift in the longitudinal position of the mode].

with respect to the wave crests was likely to occur during atest0 -

This

bad the advantage, however, that it helped to locate the position..for.

.Biari-in values of lateral force and (unstable) moment, which was found....as

an-ticipated, to coincide roughly with the position for greatest rooting of

the bow0

. . .

M.gure

b

illustratesthe fact. that in the absence of longitudinal

re-straint the ship has

a

natural tendency to take up this same. long itidtni

position, as in sketch (a), which may therefore be looked upon as a kind of

atural steady-state position possible of attaient for at least

(7)

corn-ponent of the weight resulting from the downward tilt of the water snr-face, and increased wave-making resistance forward resulting from the

rooting of the bow.

The required ship speed is easily deduced from the eression for

trochoida]. waves in deep water

V =

-'V

where v, is wave speed and X is wave length, Rewriting in the form

where is in knots and X in feet, it is seen that a ship traveling at wave speed in waves of her own length has a speed-length ratio (as

the term. is used by naval architects) of

l,3L

For a destroyer of; say.,

3S0 feet length, this means a speed of about 2S knots, or wefl. below the

usual maximum speeds of ships of this type. In still water, the

resist-ance to forward motion of a destroyer at a speed-length ratio of l,3IL is

of the order of ₅ lb./ton, or 2,5% of the weight (displacement). The forward-acting component of the weight, due to tilt, is given in sketch

(a) of Figure ii as roughly _5%. Thus there

is

evidently ample driving

force available to offset a very considerable increase of resistance caused by the rooting of the bow,

Time remainder of the present paper deals with the ship in this

steacr_state longitudinal position relative to the waves, Numerical.

values are for a 6-foot model,

IlL FORCES AND MOMENTS

FORCES AND MOMENTS IMPOSED BY FOLLOWII SEAS

It is convenient to start by attempting to estimate the magnitudes

of various forces and moments to which the ship is subjected by the

fol-TM-913

(8)

3-TM91i.

lowing seas and which are absent in still water; namely,

- the force generated by the mean inclination of the water 'sur-face (already inentionad),

- the force and moment generated

by

the local 5lope of.the wave

profile at various stations along the ship,'

'- the fo roe and moment generated by the oxbital wave velocities at various stations along the ship.

The plan-view skètchès of Figure

5,

bearing corresponding letters, illus-.trate 'these.

Effect of mean inc].i tion of the water surface.0 Neglecting for the moment the curvature of the wave profile. and considering only the

'general slope of the wavô in the regions that contribute most to the

buoy-ancy- (say, comparing stations 2 and 3 with stat.ons..7 and ,8), it isseen that the effect is as if the ship were ..resting on an inclined surface of smooth water. If the men slope of this surface, essentially represented

by

the tm 'of. the ship, be designated by '7', and the weight .of the. ship

by W, there is a forward-acting component of the weight in a direction

normal to the wave crests equal to W'T a±ad a lateral component of this

component, normal to the shipes centerline, ecijial to Wsine. For

rea-sonably small values of the' angle of heading e with respect to the normal to the vave crests, it is sufficiently accurate to let sine = e, arid to

express the transverse component of the force as

FA=WTe

(I)

as in sketch (A) of FigUre

5,

For the 6-foot mOdel, with W =

27.05 lb.

and an estimated value of 1 of'

0.05

radians

(as

in Figure 'Ii),

Lateral Force

FA -0.0214 lb,/deg.e

Effect of local wave slopes. Considering- now the actual

(9)

the ship, it becomes evident that additional forces are produced locally

by the differences of water level on-the- two sides of the-ship.-. If..dr/dx

is the local slope of the- wave surface,- measured 1ii a vertical- plane

nor-:mal to the wave crests,. then the .sloie measured in-a v-ertical plane.nor=

ma. to the ship's centerlire is

.=

(d/dx)e, where again it is sufficient

to let sine = 8.

Figure 6 shows a typical cross-section of the ship-with

the watriine sloping at this angle

(greatly exaggerated for clarity.)

Using the notation of Figure 6, considering, unit length of -the ship, and

letting w be the specific weight of water, the foUo4x

relatiOns are

seen to hold:

h1=h'+bp/2

= wh

W(h+b g/2)

_{p =:wbz w(h -b}

p/2.)

- 2 .

,w

2

P1 =-(h+ b

'2)

,.

p2 =-(h-b W2)

hb

+' (b

/2)2]

=

f[h2_hb

_p+

('b

/2)2]

Hence there is a net side force per unit length

1 -

= whb

(2)

The local forces have been computed from this expiesson for-a numberof

ship stations; and, are shown approximately to scale on slcetOh (B). of.

Figure S.

The angle p was, obtained from the equation for a trochoid,

p

(-/,)e..=

r sin

R+ r cos'

where r is the orbital radius at the water surface, .R the radius of the

gererating circle, and ?i' the angle of rotation of the generating radius.

The total force and total moment acting are evidently

Force

.=

wJ hb

dx.

0

Moment =

hh-p x dx

where £ is the length of the ship.

These integrals 'have 'been evaluated

nuineriàal]y, and the following values Obtained for the 6-fOot model:

(3)

(10)

T-,91

.6

These lateral velocities are superimposed on the forward velocity V,

LaterSi Force Moment about CG

= -0.019 lb ./deg.e MB = +0.072 ft.lb./deg.e

(C) Effect of orbital velocities. According to the trochoidaJ. wave theory, the horizOntal. component of the orbital velocity V0 of the fluid particles in a wave is related to the...speed of advance, of the-wave

the expression vjv = ,t(/X) cos ', where H is the wave height. For

Wx

1/20, .. .

= 0.157 cos

The velocity v0 is directed forward in wave crests and aft in troughs; its variation along the length of the ship is shOwn. approinate1y

in

sketch

(p) of Figure

5.

Corresponding transverse components .of velocity v =

are shown in Figure 7(a), plotted against position along the ship's length.

it is apparent that the ship is in a situation generally equivalent

to that of holding a fixed heading while in a. whirlpool0 .A reasonably good estimate of the forOe and.. moment magnitudes produced should therefore be possible by àompa.rison with' the fo:rce and moment magnitudes produced

by the rotation of the ship while turning

in

still water, for which experi-mental data are available, These data are considered in more det

il

in the next section of the paper. For present purposes it is necessaa7 to

note only that rotation of the ship results in a radial force

(corres-ponding to the coefficient Cf) that acts toward the center of the turn and a moment (corresponding to the coefficient Ck) that resIsts turning.

It will be noted that a ship rotating about its CG with an angular

velcity

= de/dt. has lateral velocities v along its length proportional to the distance x of axr..station from the CG,. so that, v xO...: In-the-.

development of the equations of motion

in

reference 1, a. space angular velocity is employed, defined as fl. = = é(21/V) where Q,

is

the length .of

the ship, and V its forward vé1octy. Figure 7(b) shows the distribution

(11)

taken in the. àhip' s centerline plane. Hence the effect is that of

de-flecting the flow 'at each point along the ship's side by -the angle v/V

(strictly tan v/V), which. therefore constitutes a local angle of attaèk. Since , in aerodynamics, lift is found to be proportional to. angle of at-.-..

tack, the lateral force on each element of the ship's side may be. expected

to be proportional to v. The velocity diagrams of Figure 7- can therefore

be considered to represent force diagrams to some scale factor K. The. total force and moment are then,.

Force

vdx

(li)

MOment = K) vxd

where, in principle, K1 = K2. The reasoning here is similar to that em-ployed by Hovgaard \/ and later

by

Smith

Before attempting to evaluate the integrals, it is necessary to...

mcdify Figure

in

one respect,

in

order to have it represent the .p1rsical

conditions correctly0 Without modification, Figure 7(b) .indicates a net force. acting away from the center of the turn; the CG -being .somewhat aft

of the mic-length of the ship., This is cQntrary to the experimental evi dence, which gives a force acting toward the center of the turn. Evidently, then, the rotation must induce an additional, outwardly-directed. lateral

velocity Vd, as indicated in Figure 8, which results in an inwaidly-di-rected force L. Reference 3 discusses the 'analor between a conventional.

model moving in a circular path of radius R, Figure 8(a), and a model. curved to the same radius R movIng in a straight path; Figure 8(b), and. concludes that the same pressures on corresponding elements, and the same total force and moment, are produced. There is abundant evidenOe from'

aerodynamic theory' and experiment that an induced lateral 'velocity_{.Vd and}

a corresponding lift L in the opposite direction are in fact produced on a curved body (say an airfoil), the chord of which lies in the direction of motions as in Figure 8(b),

The effect of the. induced velocity _{Vd can be represented on Figure} 7(b) by' shifting the datum line of the. diagram.. in a direction. to decrease

the positive values of v and increase the negative values, It was found

TM- 914

(12)

7-TM-9.i

-8-by trial that Shifting the:datwa line -8-by 0.0027 on Figure 7(b) gave results

consistent with the exerlinental . determinations, of C

and-Cj in. ttirn4.ng:.in.

5t1U water, and that the coefficients, of Equations (ii) were then in

reason-able agreement:

K1 = 1.00, K2 = 0.9. 'App]ingthe saine'shift to the dati.n

line of Figure 7(a), and using the same values of the . coefficients -K1

and

K2,, the following were obtained, for the effect of 'the orbital

velocities

on the 6-foot model:

Lateral Force

Moment about CG

Fc

-0,009'ib./deg.e

M0 = +0.07 ft.lb./deg0e

Combined effects of (A), (B)9 and.(C).

Adding the

separate.contribu-tions,. the estimated resultant force and moment magnitudes to which

the

6-foot model is subjected

by

the fo1loing seas are' seen' to be

Lateral Force

.

Moment about CG

Fk=_0002lI

. .

= -0.019

. .

M= +0.072

.

..

-0.009

= +o.OS7

',, .

Fe = -0 ,0S2 . lb

./deg .0

Me = +0.129 ft .lb

./deg.e

These are indicated on. Figure 9 by solid arrows.

IDR0DYNAMIC PROPERTI. ØF

A force and a moment are

introduced..wheneve..the-mOtioflOfaShiP43.

disturbed' from a given stear state. It is customary.to treat-separately.

the effects of disturbances in yaw and in angular

'velocity,.

In reference

1 these are defined in terms of. the following coefficients, w1ich are

(13)

Slope of the curve of lateral force vs. yaw

CL = =

Slope of the curve of moment. about the CG vs. yaw Cm = = (14/ V2At)/a'q'

Slope of the curve of lateral force vs0 spaceaiigu.lar velocity Cf

_{= acF/an=}

a(F4

v2A)/afl.

Slope of .the curve of moment about the CG vs. space angular

velocity

0k =

c/an a(W

vA2)/&c1

J

where 9' is the yaw angle,fLis the space angular vélPcity,.fL,-8! =

F is the lateral force, M the moment about the CG, A the projected side:.

(profile) area of the submerged p9rtions. of the sh.ip,L the:ship,?s length,

and p the density of the water.

-In still water with rudder amidships, the values of these coeffi-cients for the ship under consideration, ab reasonably

Small

departures

from straight-line motion, were found in earlier model tets to be

0.356

_m

= 0.069

Cf = 0.063 Ck = 0.069 For the 6-foot model, with.

L=6ft.

A = 1.162 ft.2 .

V

_{= 5.55}

ft./sec. (wave Speed.= model speed)

i=0.97

the corresponding force and moment rates for thö straight-line coeffi-.

cients C2, and C are

Lateral Force

+0.2±5 lb0/dej.'P.

Moment. abOut GG

MLp +0.250ft.ib./deg.tP

(5)

The model experiments in following seaà 'illustrated in Figures 2

and 3 were carried out in a straight-line towing tank, fitted with a wave

(14)

TM-9

-10-paddle to make transverse waves.

They covered variation of the. yaw angle,

but it will be evident that an angle of yaw 4'

involved also an angle 0

between the heading and the normal to the wave crests, in exactly equar

amowkt.

- Hence measured values of force and moment were influenced by both

q' and...Removing thein.fluence of e

by

subtracting. the estimates ot its

magnitude already discussed, the resulting values of the force and moment

due to yaw alone are

Lateral Force

Moment about CO

measured F = +0.219 1b0/deg .4' and e

M -= +O.li9l ft .Ib./deg..

4'

and 0

F0 = -0,0S2 lb./deg. e

M8.+0.129 ft.lb./deg. e;

=+0.271 lb./deg.'4)

= .+0362

ft.].b./deg.4'

Compared with the corresponding still-water values from the preceding'

para-graph1 these are seen to correspond to percentage increases of

26%

The increases can be attributed to the changes in the

longitudinal

distri-bution of the area A made by the seas, the effect of which is to reduce

the force contributions of elements near the mid-length and to increase

the force contributions of elemes' nar thS ends.

Parallel model experiments to determine the influence

of the fol

-lowing seas on the rotaiy force and moment rateS have not been attempted,

no readily. adaptable test methods being available

at present.

.. It is

reasonable tO assume, however, that since-the-rotaly rates

ist

depend-upon the distribution of the area A-tin

much the same fashion as the

straight-line rates, they will be si2thi1arly increased by the

following

seas. Under this asswnption, the rotatr coefficients become

= 0.079

= 0.100:

RUDD

FORCE AND MN

in reference 1 the force and moment introduced by.1ing the rudder

(15)

Slope of the curves of transverse force vs. rudder angle

= aC/as = a(F/

v2A)/a&

Slope of the curve of moment about the CG vs rudder angle

= aC1/a6 = a(M/ V2AL)/aS

where 6 is the rudder angle.

In still water, the values of these coefficients were ±ound in the earlier model tests previously mentioned to be

CA 0.063 = 0.029

In following seas, Figure IL shows that the rudder is near a wave

crest.. The speed of the flow across the rudder is therefore redmced by nearly the amount of the orbital velocity, or nearly in the ratio

-(V - v0)/V, and the rudder force is consequently reduced nearly in the square of this ratio: in the present instance to about 73% of the still-water force for the same rudder angle. For the 6-foot mode].,

the force and moment rates in following seas are, then Lateral Force Moment about CG

F5 = +0.028 lb,/deg.5 M5= -0.076 ft..lb./deg.5

IV. STATIC EQUE]3RItJM

It is of interest to exI1ine the conditions for static equilibrium

in the following seas because these provide a straightforward means, of

judging the adequacy or otherwise of the rudder to control the ship, apart

from a].]. crnamic considerations 'or the actual process of steering. Figure 9 indicates thedirections and senses of the static forces and moments,

where solid arrows are used for those imposed by the seas and open arrow's

for those resulting from the conditions of motion of the ship.

The equations for equilibrium (from section III) are evidently

(6) ¶rM-91L

(16)

-U-TM-914

12

-(9) F = +0.271LP + 0.0286 - 0.o52e = 0

M = +0.362W -0.0768+ 0,129e =

ô

Then these are solved for W and 6 in termà of e it is found that

= +0.011 6/0 = +i.7S (5)

The first ratio, is relatively inconsequential, but the second bringth out

the impo±tant fact that ].arge ruder angles 8 are called for to maintain equilibrium with quite moderate angles 0 between the heading and the

-normal to .the wave crests. The angle 0 is, of course, subject to inde-pendent choice and is not simply the result of accidental disturbances,

so that iii principle it may have any

a10

However, if 0 reaches 20°

the full available, rudder throw of about 3S° is seen to be required to maintain equilIbrium, leaving nothing for control or correction of

acci-dental disturbances, while if '0 exceed .20°. it evidently becomes impos-sible even.to m.ir'tain equilibrium, and the ship will 'turn (to starboard)

with full (left) rudder, out of control0

The value 0 20° thei'efore marks a limit for the ability of the

rudder to control the ship, under the assumed conditions of wave size and

ship speed.

V. DNALG CONSIDERTI0NS

EUATI0NS OF MOTION

The equations of motion can be written + - fl..(m - Cf)..= '-, ( 8,0)

nfl' - .c+nc

2(5,8)

where the

léft-hnd

sides are taken from reference 1 and the right-hand sides indicate the force aid moment due to both the rudder angle 6 and

(17)

the heading angle 0 with. respect to-theporal. to the wave- crests.

Primes

indicate. differeritationwith respect to the .nuinber..of àhip lengths traveled

S, where

S= t; m1 = M1/(A2);

rn2. = M

(M.);.ax4-n.=

/(A213..aredi-merisionless coefficients of apparent longitudinal mcs M

apparent

trans-verse mass M2, and apparent moment of inertia I, respectIve].

DYNAMIC STABUITY ON COURSE IN ST]LL WATER; RUDDER AIiIIDSHIPS

On straight course in. still water with rudder amidships

5,e)

= '2 s,e) = 0

and 'Equations (9)' are a system

of homogeneous linear differential equationi

with

constant coefficients, the solutions

of which are

where

and

b

=

(Cj/m2+

C1/n)

k

=

[cLck - (m

-

Cf)C] /'nin2

Both, exponents are real for all ships

s'ç

far studied, with P2

_{I>> I P1 I}

For the ship under consideration in still water, the various parameters

are

. . p's +

pls

'fL=fl1e

= O.3S6 0m = 0.069

oo63

'IC = 0.069

is the number of ship lengths run in time t

b

/b2

=

-+

IC rn3

0,122

m2

_=0,23S

n

= 0.0116

(10)

= -1,20 p2'= -6.28 TM-9)4

-13-p2 = -/b2 IC

(18)

r

TM-91

--'.T'hé value of p1 forms a.,convenient..indexLof stability.. -If

negative, theinotioxj is stable in the sense that .the:sl4p ret'.ii'ns-to a

now straight óourse following a small. arbitrary-4nitial. disturbance. If

p1

is

positive, the motion is unstable in the sense that the ship windà

up into a spiral (eventua3.]y - becoming a circular.) path 1'oflowing a

ini 1

arbitrary initial disturbance.

Figure 10, taken from reference 1, shows' the changes of head.ng 'in still water sibsequent to a particular disturbance for the s:ix ships of Figure 1. The destroyer (F), under consderation here,.. is seén.to settle

on a new hea4ing about 2° from the original heading, While the unstable

minesweeper (A) dOes not settle on a. new heading but continues to turn. (The. additiOn.l curve on Figure 10 'Is discussed below.')'

'MOTION IN THE FOLLOWThG SEAS; RUDD

'FID

On a straight course in the. following seas the ship Y: be considered'

to be in equilibrium prior to disturbance at ar heading angie.e. which is less than the iimitin' angle of 20° set by the rudder effectiveness, but..

which need not be zero, with rudder angle '6e and yaw angle. 4)e as

reqired

by Equations (8). :Sippôse the' rudder to remain fixed following a.disturb-ance and regard 'the quantities 4)' ani e in uations' (9) as measured from

their equilibrium values.

The' force and moment imposed by the following' seas m be written

(from Section HI):

l

=

25e'

cme

where

= ac/ae

a(F4 V2A.)/ae

=

c54

V2AL)/ae:

(19)

The equations of motion(9)then beoë

"'2 '

Ii nJ -

'P C + fl.0 = C e

the a olutions of whiôh, remembering that fL = S', are

Pis

Ps

PS

+ + = 0djJ49 0m = 00100

= 0079

= 00100

4'=9'1e

+4'2e

+P3e3

The values of the p'ts m,r be found from the condition that for a non-vanishing solution the determinant of the c oeffic iénts vanishes0 This

leads to the cubic equation in p

+ bpi +cpi + d = 0

where

i isl,2,or3

b =(C/m2 + C/n)

[Ck._(m_C) Pm"'':

d ₌ _{[Cm C2,} _{- C Cm] i'}

For the ship under consideration in the following seas,

= -0.086 e

C = 00036

me

The positive value of p1 means that in the following seas the ship, which

is crnainicaJJ,y stable in still water, behaves like a dynamically unstable ship. Thus an initial disturbance will cause an. exponentially-increasing

divergence in heading. 0.122 = 0.23 n

= oou6

p1. p2 = -1.92 p3 = -9.11 TM-924

(20)

TM-91L -

16-The additiona]. curve On Figure 10 shows the changes of heading .e

calc'A1ated from the first of EquationS (13), anm1 ng the same initia].

disturbance as or the still-Water curves. It will be seen that the-

ap-parent inStability of the ship in the following seas is much the same up to about 6° change of heading as the instability of the minesweeper (A) in still water, but becomes rapidly more pronounced at greater changes of

heading. The point was made in section I of this paper that as a matter

of practical experience the minesweeper (A) cpn be successfully steerd

by hand in ordizary weather. It may be concluded, therefore., that the destroyer (F)

can

be successfilly steered by hrr1 in the following seas.

However, the minesweeper '(A) in still water requi.res constant attention

and prompt responses on the part of the helmsman; this must be t±ue also of the destroyer (F) in the foUowg seas, with the additioha]. proviso

that moie rapid responses will be needed if' the ship iS allowed to swing

too far before checking.

VI.

CQLUDING .1ARKS

Two effects of following seas have been considered for a particular ship under specified

conditions:

1) In' section IV it was shown that the full available. idder-tra.ve1 of _3S° is needed. to maintain 'eqüilibriumwheñ the desired course with

respect to a normal to the wave crests is 200. ,'

a) hi section V it was shown that the ship, which has th.e.greatest rni{à

stability in

still water of ar of the six ships of Figure 1., be-haves

in the

following

seas

much like the

most

unstable ship of Figure 1

in still water. .

The

first of

these effect'è appears .to be the more important ol' the two. A restriction on the ma,d.mum permissible course to aztbing like as little as 20° to a normal to the wave crests is in itself stringent enough. In practice, hàwever, the course must be kept well within the calculated

(21)

1 imlting !alue. because some part pf 'th

ávailabIe ruddei travel must be

held. in reserve to .correc± accidental distubances Si'id. iii pàrticilar-to

check ary angular velocity resulting from a disturbance0

Besides, sea

waves are seldom regular enough to allow vex

accurate 'estImates to be

:made of the direction of their travel.

'

Still water imposes no great requirement on the rudder needed, to

steer a stabl

ship. Angular velocities pi'oduced br accidental,

disturb-ances die out of thselves, without application of the.rudd'er.,. and the.

resulting small 'changes of heading could be corrected for slowly, with a

rudder of verymodest size,

Following seas, on the other hand, evidently

impose a'stiff requirement on the rudder, the effectiveness of which may.

easily become the governing factor limiting

,

The' usual criterion

for fidng the necessary rudder effectiveness is the desired minimum

turn-ing radius with hard-over helm, in still, water, .A secbnd criterion; might

well be the ability to steer in following seas of specified size without

reducing speed..

- . . .

-The second effect 'of the following seas, namely, the apparent loss

of stability, with its implication of' greater. difficulty in. steering and

the consequent difficulty of preventing the heading -'angIe- from -exceeding

its limiting value uhless corrective rudder applications are made very

promptly, is of 'special interest because of its magnittide.

For the

as-sumed conditions, the apparent loss of stability evidently corresponds

in effect to a shift of the p1 value in still water all the way across

the range of values of this index embraced by Figure-i.., It may be. argued

that under these circumstances the precise.p1 value in .stillvrater, to

within a close tolerance, is of little consequence.' On the other hand, a

study of the governing equations indicates that in general an increase in

the negative p1 value in still water will diminish the apparent instability

in specified following seas.

In conc].usion, it may be pointed out-that the particular. combination

of wave size, ship speed and ship attitu4e adopted for the numerical work

of this paper 'is adinittecfly an.. extreme one

_{It is not intended, therefore,}

TM-914

(22)

-17--18..

that broad conciuions regarding desirable' ship characteristics should. be

drawn from the numerical results;. such .cthiclusions would be premature.

It is of interest., nevertheless, that the. numerical Tesults

ndicate. the

possibility of broacig-tç under the extreme conditions. thr: represent

This is believed to be reasorab]

consistent With practical :experience'

over the years with ships of all types, and, to constitute,. in a sense, a

calibration of this practical experience in quantitative. term3.

,Acidiow1édhent is made of the' important help of Professor

B.V. Korvin-Kroukovsir of the Experimental Towing Tank, Stevens

(23)

Davidson, Knneth S.M. and Schiff, Leonard L:: "Turning and:

Course-Keeping Qualities".

Transactions of the Society of Naval

Architects. and Marine Engineers, VOl.

SIi, l9l6, pp. l2-2OO.,

Hovgaard, W.:

"Turning Circles".

Transactions, of the Thstitution

of Naval Architects,

Vol.'hh, '1912.

3°

Smith, Richard H0:

"Curvilinear Dynamics of Airships based on

Bowed Modal Tests".. Proceedings of the Fifth Interiational

'Congress 'for Applied Mechanics, Cambridge, Mass.

,

(24)

-.19-TM. 914

20-Figure 1 Tur24ng ability vs. dynamic stability index p1 for various ships and other bodies.

Figure 2 Mode]. test of the Destroyer (1) in waves of length equal io 3/ of the ship' a length and wave height/length ratio of 1/20. Figure 3 Model test of the Destroyer (F) in waves of length equal to

-1-]j'2 of the ship's length, and' wave height/length ratio of 1/20.

Figure ii Destroyer rwming at wave' speed in waves of her own length.

Figure Effect of the following sea on lateral forces and yawi 'momentst

A) force generated by the mean inclination of the' water

surface, ' ' '

-) local lateral forces generated by transverse components

'of the local slope of the wave profile at various ship

stations,

0) orbital velocities of Water at various stations along the ship's length.

Figure 6 Lateral force produced

by

the' local slope of the wave prpfile at a station along the ship's length.

Figure 7 Distribution of the lateral velocity comonent: - a) due to

the-orbital Wave velocities shawn onFig.S(C),afld'b).dUetO a space angular velocity

ft

= 0.005 in turning; Velocities

in

ft../sec.

Figure 8 Càmparison of the motion of a ship!S'

huh

in two cases: a straight model in a circular path of radius R, and

a mOdel 'bent to the radius R in' a straight path, Equal

Figure

LIST OF APTINS

lateral forces L, and induced velocities Vd are produced.

.9 Equilib±'iumof the static forces and'moments'acting on a:ship,'s

ha].]. in following seas: solid' arrows indicate the forces and' moments inposed by the waves;: open arrows indicat'e 'the

reactions-due to the rudder angle 6 and the 1rdrodynamnic properties ,of

(25)

Figure 10 Change - of heading 6 ys, distance run a, in ngths, for the

six ships of Fig. t in still water with rtidder amidships,.

-and for the Destroyer (F) in the following, sea with ru.dder

fixed. Initial disturbance 10 yaw with no angular velocity.

TM-91

(26)

21-I

3flOId

0 P

(D/L)min

X RUDDER EFFECTIVENESS 1%) A3S3P M3N 'N3)I080H A9OONHO3.L dO 31fl.LILSNI SP1BA3J.S

)(NL

ONIMOJ.

1ViN3I4I3dX3

\

MINESWEEPER (A)

z

'U)

I

I!

rn

.-

(0 BATTLESHIP(C) 1U) 1-4 I

Ir

w -1 -c

o

-n -I

0 \

TANKER (D)

-

c

D

P1 CRUISER CE) T1 rn U) DESTROYER (B >c

C

0

DESTROYER(F)

z

H

p

I-'

p

ro

(27)

-n

Q

C

n

FIGURE 2

_{EQUAL TO 3/4 OF THE SHIP'S LENGTH AND WAVE HEIGHT/LENGTH}

MODEL TEST OF THE DESTROYER (F) IN WAVES OF LENGTH

RATIO OF 1/20.

I P

p

(28)

F 0) ___,pt1

)W1

S

0

I'

- I)t; '

-

,----

---a.

'1

FIGURE 3

MODEL TEST OF THE DESTROYER (F) IN WAVES OF LENGTH

P1

_{EQUAL TO I'/2}

OF THE SHIP'S LENGTH AND WAVE HEIGHT/LENGTH

RATIO OF 1/20.

r

(29)

FIGURE 4.

_{SHIP RUNN!NG AT WAVE SPEED IN WAVES OF HER OWN LENGTH.}

IO%OFA

_{WT :5%OFA}

SKETCH (c), SHIP SLIDING FORWARD

SKETCH Id), SHIP SLIDINGBACKWARD

):J1

_{;:- s!:i34+:2O}

SKETCH (a), SHIP IN NATURAL STEADYSTATE"

_POSITION

SHIP TRAVEL

SKETCH (b)1 SHIP INMUNSTEADY"POSITION

_{WITH WAVE CREST AT C.G.}

Figure 4

Deetrcyer running at wave speed in waves of her oun 1gth.

WAVE TRAVEL

p

lILT BY THE HEAD ABOUT 5% OF 1.

(30)

Figure Effect of the fo11oing. eea o lateral forces and yasing _mentàs force gecerated by the mean incI4-ntion of the water surface; local lateral forces generated by transverse components of th. local elope of the wave profile at various ship stations, orbital velocities of water at variOuà stations along the.

length.

(31)

Figure 6 _{Lateral force produced by th looá3. elope of} _{the waveprcfile} at a etation along the ehip'e length.

(32)

ci

-o

0092,

.0130

-0.0152

7,

/

+0

.0152

CG

TURN TO PORT

- 0.0092

.+ 0.0148

0.0027

1

5. 4,'3.

a),

(b)

Figure 7

_{Distribution of the lateral velocity components: a) due to}

the orbital 'wave velocities shown on Fig, 5(C), and b) due

to a space azgular velocity fl = 0.005 in turning. Velocities

1n tte/aec.

FIGURE 7

0.0027

0'

I-,

_{ANG EOTOSTD..}

f

0

(33)

a--n

C

In

Figure 8

Compariaon

of the motion

of a ehip's hull in two caaesz

aatraightmodej

inacircularpathof radius R ,and

amodelbenttotheradiva

Rinaetratghtpath. Eual

(34)

Figure 9 quj14bzmofthe etatic forces and momenta actiog on a ship's

hull in following seas z solid arrows indicate the forces and

moments imposed by the waves; open arrows indicate the reactions due to the rudder