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R 945

DAVIDSONLABORATORY

Report 945

METHOD OF PREDICTINGCOURSE STABILITY AND

TURNING QUALITIES OF SHIPS

byWinnifred R. Jacobs

U 0 l 0 GMarch 1963

II

DAVIDSON LABORATORY[ REPORT 945

[1 March 1963

METHOD OF PREDICTING COURSE STABILITYAND TURNING QUALITIES OF SHIPS

by

Winnifred R. Jacobs

Prepared under sponsorship ofOffice of Naval Research

Contract Nonr 263(40)Technically administered byDavid Taylor Model Basin

DL Project LZ2388/041

Reproduction in whole or In part is permittedfor any purpose of the United States Government

Approved

S. Tsakonas. HeadFluid DynamicsII,

I

I" ABSTRACT

An analytical method combining simplified potential

flow theory and low aspect-ratio wing theory with empirical

modifications for a real viscous fluid is used to predict the

stability derivatives (first order hydrodynamic force and

moment derivatives) of a family of hulls in order to estimate

the dependence on geometric characteristics of course stability

I and turning or steering qualities. The hulls are Taylor

Standard Series forms with after deadwood removed and have

the same length and prismatic coefficient but varying length-

draft and beam-draft ratios and skeg area. Comparison betweenthe values calculated by this method and those obtained from

experimental measurements shows good agreement. The analytical

method can predict the relative effects of the geometrical

characteristics. Calculated magnitudes are slightly differentfrom the experimental but are on the conservative side. How-

ever, since the hulls tested have the same prismatic, the em-

pirical modification for the rotary moment derivative which is

a function of prismatic coefficient has not been fully tested.

Necessary refinements of the method must wait on analysis of

data on series of forms with different prismatic as well as

Iilength-draft and beam-draft ratios.

R9

L7

SR-94!5

NOMENCLATURE

A profile area of wing or hull, ft 2

AR aspect ratio of wing

B beam, ft

CL dimensionless lift coefficient based onprofile area

Cs two-dimensional lateral added mass coef-ficient (sectional inertia coefficient)

D diameter of turning circle, ftRf + R

D- f r total resistance coefficient of the hull

F force, lb

Ft = Fy measured lateral force coefficient

g acceleration of gravity

H maximum draft, ft

h local draft, ft

hT maximum skeg height, ft

10 moment of inertia of hull, lb-ft-sec2

I added moment of inertia of entrained waterz (see text), lb-ft-sec2

ki

k2 Lamb's coefficients of accession to inertia,longitudinal, lateral and rotational

k 'f

L lift, lb

L' = L dimensionless lift coefficient based onP 2.tH area t.. H

It length, ftA

M = mass of hull, slugs

R-945-iv-

mI 02 hull mass coefficient

mI k km? longitudinal added mass coefficient

2 mI lateral added mass coefficient (see text)

1 m , m + M• longitudinal virtual mass coefficient

!my mt + mL lateral virtual mass coefficient

i ' rotational added mass coefficient (see text)

N yawing moment, lb-ft

SNt N dimensionless yawing moment coefficientI2 t2H

n' 0 z virtual moment of inertia coefficient

iR radius of turning circle, ft

R f frictional resistance, lb

SR rresidual resistance, lb

i r' = dimensionless angular velocity

S Ut dimensionless distance along the path of-7- the center of gravity of the hull

t time, seconds

I U velocity of the center of gravity .of thehull, ft/sec

u x = U Cos x-component of U, ft/sec

Iu Y= -U sin y-component of U, ft/sec

Sx, y, Z coordinate axes fixed in the hull withorigin at the center of gravity

x distance from LCG of center of gravity ofSlateral added mass, ft

Xp distance from LCG of center of pressure ofp lateral force Y, ft

X xT distance from LCG of center of pressure of

tail surface or skeg, ft

R-945I -V-

xs, Xb x-coordinates of stern and bow, respectively

Y lateral hydrodynamic force, lb

Y' = lateral hydrodynamic force coefficient

(3yaw angle or drift angle6 rudder angle

A displacement of hull, lb

p mass density of the fluid, slugs/ft 3

01,2 stability indices

Subscripts (other than those in above definitions)

H refers to bare hull

I refers to ideal fluid

r' refers to derivative with respect to r'

T refers to tail surface or skeg

/3refers to derivative with respect to

R-945-- Vi--

!

TABLE OF CONTENTS

Page

j Abstract iii

Nomenclature iv

Introduction 1

The Analytical Method 3

I Simplified Flow Theory 3

The Assumed Static Force and Moment Ratesfor the Bare Hull 7The Assumed Rotary Force and Moment Ratesfor the Bare Hull 9

3 The Stability Derivatives for Hulls with Skegs 11

The Stability Indices 13

Presentation and Discussion of Results 15

Conclusions and Recommendations 18

1 References 19

I Table I

Figures 1-12

I Appendix - Data Charts

I

I

R-945

i-vii-

INTRODUCTION

The object of the program whose results are re-

ported here was to develop an analytic method for estimating

the course stability and turning or steering qualities of

I" ships. The method, following Martin,' was to be based on a

combination of simplified flow theory and low aspect-ratio

wing theory and was to be assayed by comparison with signifi-

cant empirical results.

Measurements exist of lateral force and moment, on

straight course and in turn (rotating arm tests), on a seriesof eight models (the "840" Series) having the same parent as

I! the Taylor Standard Series without the deadwood (faired-in

skeg) aft (Fig. 1). The models were of the same length and

[ prismatic but with varying beam and/or draft and hence dis-

placement. In addition, there are measurements taken on the

[ models with various flat plate skegs added (Fig. 2), but no

other appendages.

[ Data measured in 1946-47 were reported in refs. 2

and 3 which were concerned primarily with equilibrium turn-[ ing conditions. Measurements were limited in some cases to

a very narrow range of yaw angle around the equilibrium turn-

ing angle for a given diameter of turn. For this reason, the

data on models with skegs reported in ref. 3 are useless forthe purpose of this report. However, unpublished experimental

I data over a wide range of yaw angle at several turning radii,obtained in 1951 at this laboratory, are available for two of

the models with and without a plate skeg extending to the

stern. A third hull was tested here in 1959, without skeg

and with skegs of various sizes, and an analysis of the data

was reported by Tsakonas. 4

I Martin' recently reanalyzed the straight coursedata obtained in 1946-47 on the eight bare hulls along the

R-945

-'-__ _

lines suggested by Thieme,5 Inoue6 and Fedyaevsky and

Sobolev. 7 His method of treating the problem of ship motions

in the horizontal plane is based on a modified low aspect-

ratio wing theory, the Munk ideal moment and cross-flow drag

theory. The last was used in estimating the nonlinear force

and moment range.

The present work considers only stability deriva-

tives of the first order so that the nonlinear variations,

the quadratic and higher terms, may be neglected. Also,

since in the linear range the straight-course data are much

sparser than the rotating-arm data, it was decided to give

more weight to the latter in estimating static stability de-

rivatives. It has been the experience at Davidson Laboratory

that entirely reliable static force and moment rates for

straight-course motion can be obtained from rotating-arm data

at sufficiently large turning radii. This was confirmed in

ref. 4 where rotating-arm results were compared with straight-

course data measured in Tank No. 3 of Davidson Laboratory

using the same towing system and measuring devices as for the

rotating-arm experiments in Tank No. 2. In this way, errors

attributable to inconsistent mechanisms were avoided. The

earlier straight-course data had been measured in Tank No. 1

using different towing and measuring apparatus.

Landweber and Johnson8 have shown that the simpler

methods derived from an alliance of potential flow theory

and low aspect-ratio wing theory can estimate the stability

derivatives as accurately as sophisticated methods based on

more realistic theoretical considerations. The more complex

methods employed in ref. 8 require as many simplifying

assumptions and empirical correction factors if not of the

same kind.

The analytical method adopted here combines the

force and moment approximations which were derived for

R-945

-2--

I

spheroids from potential flow theory by Lambe and for long

I slender bodies with tapered or pointed ends by Breslin' 0

with Albring's" empirical modifications for viscid flow.

[ This approximate method is based on simple concepts, yet

correlation can be considered good between calculated and

Ii experimentally obtained values. Necessary refinements must

wait on the availability of further data on hulls of other

[ prismatic with and without skegs.

This project was sponsored by the Office of Naval

Research under Contract Nonr 263(40) and technically ad-

ministered by David Taylor Model Basin.

I THE ANALYTICAL METHOD

I Simplified Flow Theory

In the potential flow theory the hydrodynamic force

[ and moment rate coefficients, or stability derivatives, of an

elongated body of revolution without appendages are defined

for the linearized region of small angles of attack and large

radii of rotation as:on straight course, r' = t = 0

I H= Y = 0

N' = m' - ml N' (Munk ideal moment)2H i

in turn, around 0 0, (1)

YrA =0

HNr•t=

The notation is that of the Society of Naval Architects and

Marine Engineers (see Nomenclature and Fig. 3). The measured

L lateral force coefficient is defined asF' = Y' - (m' + ml) r'

0y

R-945

-3-

and its derivative with respect to r' as

6F1y Yr (m' + ml) (2)7r- r' ml)

where m' is the mass coefficient of the hull and ml is the0

longitudinal added mass coefficient. Lamb, 9 considering the

added mass term as a hydrodynamic force, defines

Y H m1 k ml where k, is the coefficient of longi-

tudinal accession to inertia. Equations 1 are equivalent to

those derived by Breslin' 0 for a long slender body with

tapered or pointed ends from three-dimensional singularity

distributions.

The first of eqs. 1 is known to be in serious error

in a viscous fluid. To quote Arnstein and Klemperer: "When

an airship is propelled at an angle of attack, lift forces

are created in a similar manner as by the wing of an airplane.

It is true that the airship's shape as a wing is very poor

and its aspect ratio extremely small; but the size of the ex-

posed surface is so great that tremendous aerodynamic force

components at right angles to the flight path can be evoked."', 2

Fedyaevsky and Sobolev 7 have defined the forces and

moments acting on a ship by identifying the body of the ship

with a wing. In this analogy the span of the wing is assumed

to be double the draft of the ship to take into account the

action of the free water surface. Tsakonas 4 shows, by a com-

parison with the wind-tunnel results of Flax and Lawrence,13

that this "solid wall" method of accounting for the free sur-

face effect is correct for moderate speeds when the influence

of wave making may be neglected.

Albring" has derived approximate formulas for the

stability derivatives of a body of revolution with and without

appendages moving in a viscous and eddying fluid. The lift L

on a bare hull is the force developed by the "imagined angle of

R-945

-4-

II

of attack at the stern" of a "correspondingly shaped solid

[ without the effect of curvature." It is a3sumed or taken

from experimental measurements. It acts at a distance xp

[ from the center of gravity and Albring suggests that for most

bodies of revolution Xp = 0 is a good approximation. For a

I bare hull his results are given in ref. 11 as:

on straight course, r' - = 0,

I L' assumed or measured

I Y' = L' + D

% LH (3)

I N' m'- m•l + '•m- ml ft N'i

[ in turn, around • = 0

xY, I -- L' 0o

r H 1H

Albring does not give a formula for N? for the

body without appendages. If one assumes that the center of

pressure does not shift when going from rectilinear to curvi-

linear motion, then

I N = - (J)2 L'•H;t 0 (3a)

I On the other hand, on the basis of Albring's assumption that

the center of pressure is unaltered by the addition of fins

[or skegs, his formula for N1 r' for the body with fins should

hold for the bare hull:

Ix Nr' (X)2Nrl, L - L(3b)

R-945

-5-

x

where 7is implied by Albring to be half the prismatic coef-

ficient of the body. Lh in this case would be the value for

the hull alone, L' since the effects of the appendages arePH'

taken as simply additive.

Equations 3 are derived from simplified flow theory

with the necessary condition that in viscid flow the drag is

not zero, and so at an angle of attack the lift force is not

zero. Under this condition there is an additional moment due

to the action of the lift force at a center of pressure dis-

placed from the CG by x 0 O. In rotation, the angle of attack

at the center of pressure is changed by an amount

tan-1 (- r) _-x-- x -

where

r angular velocity

tr -

U = forward velocity

= length of the hull

R = radius of turn

The additional force coefficient resulting from this change

in angle of attack isx 1

and therefore the rotary force derivative of eq. 1 has an

additional term

_xL

which is equivalent to the negative of the additional static

moment derivative.

R-945

-6-

IIn treating the long body equipped with fins, skegs

[ or control surfaces, simplified theory assumes that there is

no interference between the body and these surfaces so that

[ their separate effects are additive. Tail-surface (skeg)

effects would be derived from the lift on the surface

(obtained either by measurements or by assuming the surface

to be a wing) by following the reasoning of the previous

paragraph. Then

L' assumed or measuredAT

NT T L(T4 _ ( 4 )

Nr,t = - -rT•

N , (7-) L1

i where xT is the algebraic distance between the center of

gravity and the center of pressure on the tail (XT isnegative).I

Equations 3 and 4 are the basis for the calculation

method used in the present report. The assumed lift rates and

the other force and moment rates derived from them will be

discussed in the following sections.

I The Assumed Static Force And Moment

Rates For The Bare Hull

[ The derivative of the lift coefficient with respect

to AL , is assumed as given by Jones' formula for a low

aspect-ratio wing of span equal to twice the ship draft. The

dimensionless lift rate per unit wing area

R-945

ii -7-

6CL = 7 AR

is derived from the consideration of elliptic load distribu-

tions along the chord and the span of a thin foil. Nondimen-

sionalized on the basis of the area H x t, the static ratesof eq. 3 become

7HTH

LI ZrH N r 1(5)y,- + DI

% 0

NI Nhi + -fLý 2~ 1m +PH i H

where DI is the drag coefficient obtained from the TaylorStandard Series curves of resistance 1 4 and xp is taken as thedistance of the center of area of the bare hull profile fromthe center of gravity.

Tsakonas 4 and Martin1 found the Jones formula to be

a good approximation of the static lift rate of the StandardSeries hulls. The fact that the formula does not predict thevalues obtained from measurements on flat plates of the sameprofile does not detract from its usefulness for cambered

wings. Bisplinghoff proves that an elliptically loaded wingmust have an elliptic planform and he points out that it isthe so-called flat-plate chordwise distribution which "compares

very favorably with the measured chordwise distribution ofpressure difference over a slightly inclined, thin, uncambered

airfoil developing the same lift. The only significant dis-crepancies come from within a few per cent chord lengths of

the singularity [at the leading edge]."'" Tsakonas 4 referredto Crabtreelsle work in substantiation of this statement.Crabtree had found that the pressure distribution over a thin

R-945-8-

plate (less than 12% thickness to chord ratio) at various

incidences showed a pronounced suction peak near the leading

edge with a consequent steep adverse pressure gradient and

j laminar boundary layer separation. Since the size and form

of the separation region or "bubble" has a large effect on

the lift, Tsakonas cautioned against use of the experimental

measurements on flat plates to predict forces in the case of

the ship-wing analogy.

The Assumed Rotary Force And MomentIiRates For The Bare Hull

From simplified flow theory the rotary force de-

rivative for a body of revolution without appendages is zero

if the body has fore-and-aft symmetry. For a bare ship hull

without fore-and-aft symmetry, on the other hand, eq. 3 shows

that:

= - H Ni - NIiH hH 'i

[and the measured force derivative with respect to r' (eq. 2):

IF 'H in i'=Y _ (m + m1) - L

where xp is different from zero although small.

From eq. 3a:

=r I LAH A

[which is very small. In addition, as shown by Martin,' there

is a term due to the asymmetry of the rotary added mass of en-trained water, m'. The moment due to this added mass is

Z"-mtr' 5/0 where x is the distance from the center of gravity

of the hull to the CG of the lateral added mass.

R-9 5'U

_ _ _ _ _

Then eq. 3a becomes

Nt mv L(6a)rHz ýH

If eq. 3b is used

N11= -m L? (6b)rHz T r H

where x/I is half the prismatic coefficient. The values ob-tained from eq. 6b have been found to be much closer to ex-

perimental results than those calculated by eq. 6a. The

second term of eq. 6b is approximately 1-1/2 times the first.For this reason, eq. 6b has been taken as the assumed rotarymoment rate. As in ref. 1, m' and 3 have been estimated inzthe following way:

kc' = k k' m2mz7= 2 m! 2 p-

2

xb

m2 = k2 0Cs h 2 dxSs

k2, k' are Lamb's coefficients of accession to inertia,lateral and rotational.

Xs,Xb are the x coordinates of the stern, bow. Jh = local draft

Cs = two-dimensional lateral added mass coefficient.It is determined at each section from the curveson two-dimensional forms of Lewis' sections byProhaska. 17

R-945

-10- I

I

xbx = LCG SX

s

1.The Stability Derivatives

For Hulls With Skegs

[ The force and moment derivatives for the hulls with

skegs are taken as a simple addition of the corresponding

[ forms of eqs. 6 and 4:

L- = L' + L'PH AT

SY= Y' + L'1B H AT

N' = N + (7)A hH -- 4

XT

Nr, -N' L'

I In the present nondimensionalized form of Jones' formula

r L'AT = i (8)

hT = maximum skeg height

The center of pressure of the skeg will be assumed at the

after end of the skeg. Since the length of the skeg is small

in comparison with the length of the hull, any error involved

in this assumption will be small.

IR9I ~R-9k5i

-1i- I

For the case of models with skegs extending to the

stern (Fig. 2), these formulas give essentially the same re-

sults as those obtained by the method suggested by Martin. 1

He included the following terms in his force and moment

equations, respectively, to account for the skeg effect:

- m(xs)ux (k.uy + k'r xs) (from eq. 2 of ref. 1 for Fy)

(9a)and

- xs m(xs)ux (k uy + k'r xs) (from eq. 3 of ref. 1 for N)

(9b)where

m(x s= two-dimensional lateral added mass at the sternper ft

ux = U cos 3 U

uy -U sin 3P -Up

r = angular velocity of the ship.

Yhen the section at the stern is that of a flat plate, ,

m(x )=r-

(see Kuerti, McFadden and Shanks 1 8 ) Iand this term is numerically the same as that given by the

Jones formula for a low aspect-ratio wing of span equal to

twice the ship draft. After nondimensionalizing, since

k2 kI 1, eq. 9a for the added force is approximately

L'(/3 - r' q-~)IAT

and eq. 9b for the added moment is IA T XT)

R-945

-12- 1

where L' is given by eq. 8. Then the force and moment rates

are those given in eq. 4.

j The Stability Indices

The criteria for inherent dynamic stability of a

free body moving on straight course in the horizontal plane

are the damping exponents a. and 02 in the solutionI o.s 025 Oj.s cOaS

1 = le + A2 e , r'= r' e + r' e/3=3e1 2

of the homogeneous linearized equations of motionis

(m' -Y' ) r' - m' As - Y, A =0x r y A (10)

n' r' -N',r - N = 0z r

where Y,, Nb, Y ,.and Nr, are defined in eqs. 6 and 6b for

the bare hull and in eq. 7 for the hull with skegs.

0 10 +

I H

Ian.0 m8

S of (assuming the radius of gyration is equal to ot

I4

xb

Iz =k' C h X~d (ref. 1)

Ii ~and

SPs =_ L •r' Uta s, rs =s- s = -'F

SSolution of the characteristic equation of eq. 10 gives the roots :

R-945

-13-I- n Ifo

-(nztY'-mý')' )±(nz'Yf-m'Nrt,)2+4nztmy [N't, •" x ' °,)N

2n' m 2n'

If al and a2 (or their real parts) are both negative,

the motion is inherently stable in that an initial disturbance

damps out exponentially; the more negative the exponents, the

sooner it damps out and hence the greater the stability. If

a2 is positive, the motion is unstable and the hull cannot

keep to a straight course without application of a corrective

rudder.

The turning characteristics of a hull in turns that

are not too tight (when nonlinearities can be neglected) can

be predicted qualitatively from the al index. A more dynam-

ically stable hull will turn in a larger radius under a given

applied rudder force than will a less stable hull. Conversely,

the more stable hull will require greater rudder force than

the less stable hull to turn in a given radius. On the other

hand, an unstable body may turn in the opposite direction to

that indicated by the applied rudder and will need a com-

paratively large force to bring it around (until it becomes

stable in the turn, but stability in turn is outside the scope

of this report).

R-945-l':- I

I

PRESENTATION AND DISCUSSION OF RESULTS

The experimental data measured in 1946-47,2I3 1951

(unpublished), and 1959, are shown in Figs. A-1 to A-48 in

the Appendix, plotted in dimensionless form versus dimension-

loss angular velocity r' = -./R and yaw angle P. Figures

A-1 to A-19 present the measured lateral force coefficientsF' = -(mo + m•) r' + T' and Figs. A-20 to A-38 the measured

y

yawing moment coefficients N' for the bare hulls. Figs. A-39

to A-48 show these quantities for three of the hulls equipped

[ with skegs but no other appendages. Skeg 20 or A is the full

skeg extending to the after perpendicular (Sta. 20), skeg 18

[ or B extends to Sta. 18, 0.1 length from the after perpendi-

cular, and skeg 17 or C to Sta. 17, 0.15 length from the after

[ perpendicular.

The experiments had been conducted at speeds ranging

[ from length-Froude numbers of 0.16 to 0.23, for which range

the implication of the analytical method that wave making may

be neglected is valid. In that speed range the ratio of force

or moment to the square of speed changes only slightly so that

data measured at different speeds within the range can be com-

I pared legitimately.

Table I gives the pertinent characteristics of the

eight hulls and the necessary information for the calculations.

The resistance coefficients Do = 2 (Rf + Rr)/pU2 H were taken

I from the Taylor Standard Series curves 1 4 as the average of the

slightly different values for Froude numbers 0.16 and 0.23.

[ The calculated Y'A and NIP, force and moment coef-

ficients respectively at r' = 0, are also shown on the data

charts in the Appendix. The data plotted at various A versus

r' are faired to the calculated values at r' = 0 with no

[ stretch of the imagination. The static force and moment rate

coefficients are the same whether predicted by the analytical[ method or by rotating-arm data.

R-945

1-15-

Figures 4 through 9 in the text are summary charts

comparing experimentally obtained values for the rotary as

well as static derivatives with those calculated by the

analytical method. These values are shown for varying length-

draft &/H and beam-draft B/H ratios.

The calculated )F'/ r', Nr, and stability indices

a1,2 predict the variations with M/H and B/H correctly but

underestimate the actual magnitudes slightly in the bare-hull

cases, more in the cases of hulls with skegs. The quantita-

tive predictions, however, are on the conservative side.

These comparisons suggest that the simple method adopted here

can be useful for estimating the stability of a given vessel.

The good results Justify use of the ship-wing analogy and the

Jones formula for the lift on the ship as a low aspect-ratio

wing.

The following are specific deductions from the

charts:

1. The static force derivative YV varies inverselywith L/H (or directly with aspect ratio) and directly with

B/H although B/H has very slight effect. Y? is increased by

adding skeg area. An increase in YV is in the direction ofgreater stability.

2. The static moment derivative Nb also varies in-

versely with t/H and directly with B/H. This destablizing

moment rate is reduced mainly by increasing .,/H and by adding

skeg area at the stern.

3. The rotary force derivative 6FI/ r' becomes 3less negative or more positive by increasing t/H, by de-

creasing B/H and by increasing skeg area at the stern. As in

the case of NI, increasing 4,/H or decreasing B/H is in the

direction of greater stability. The variation with B/H arises

from the variation in the longitudinal virtual mass coeffi-

R-945

-16- I

cient. This is the most important effect for the bare hulls,

and is as important as the skeg effect in the case of hull

with full skeg.

14. The rotary moment derivative N', is independ-

ent of B/H. It becomes less negative with increasing t/H

and more negative (towards greater stability) when skeg area

is added at the stern. The variation with tL/H is slight in

1. the case of the bare hulls, more pronounced for the hulls

with skegs.

[5. The stability indices a122 which combine the

effects of static and rotary force and moment rates, show

that stability depends almost entirely on B/H and very little

on t/H in the range tested. Stability increases as B/H de-

creases. On the other hand, an increase in B/H would result

in greater turning ability.

Figure 10 is a summary chart comparing the calcu-

lated Y', N1, Y',, N', and a, with the values from measure-ments obtained by Tsakonas 4 in 1959 on the 842 hull, without

skeg and with three skegs of different sizes. The calculatedstatic derivatives Y' and N' are identical with those obtained

from experimental results. This is also shown in Figs. 4 and7. Figure 10 shows that while the calculated and experimental

j magnitudes of the rotary derivatives and stability index differ

slightly, the stability predictions are conservative estimates.

[ It is also seen that the analytical method can predict the

trend in stability with increase in skeg area. Figures 11 and

12 for the hitherto unpublished 1951 data confirm these con-

clusions.

I

IR-945

LI -17-

CONCLUSIONS AND RECOMMENDATIONS

An analytical method for estimating course stability

and turning qualities of ships has been developed and com-

pared with available experimental data on a series of eight

hulls, the 840 Series, of the same length and prismatic but

varying in draft and/or beam and displacement. The analyti-

cal method combines simplified flow theory with low aspect-

ratio wing theory and makes use of Albring's empirical modi-

fication for the rotary moment derivative.

Encouragingly good correlation is shown between the

calculations by this method and the results based on the ex-

perimental data. However, since the 840 Series is a family

of hulls of the same prismatic, Albring's modification for

the rotary moment rate which is a function of prismatic coef-

ficient has not been fully tested. Necessary refinements of

the method must wait on analysis of data on hulls of other

prismatic, with and without skegs or deadwood aft.

It is recommended therefore that all available data

on hydrodynamic forces and moments, in turn, for other hullforms with different prismatic as well as length-draft andbeam-draft ratios be assembled and analyzed with a view to

checking and refining the method. It would also be advisable

to do further experimentation in still water on families ofhulls such as the Series 60. The latter series has been the

subject of extensive tests to determine resistance, bending Imoments and sea-keeping qualities in waves but the stability

on straight course and behavior in turn have so far been in-

vestiga~ed for only one of the series. 2 0

R-945

-18-

i REFERENCES

1. Martin, M.: "Analysis of Lateral Force and Moment Causedby Yaw During Ship Turning," DL Report 792, March 1961.

2. Sutherland, W. H.: "Progress Report on Related-ModelTests for Turning Studies: Effect of Changes of Beam and

Draft of Bare-Hull Models," ETT Report 338, January 1948.

3. Sutherland, W. H.: "Progress Report on Related-ModelTests for Turning Studies: Effects of Plate Skegs Fittedon Bare-Hull Models," ETT Report 346, April 1948.

4. Tsakonas, S.: "Effect of Appendage and Hull Form on theHydrodynamic Coefficients of Surface Ships," DL Report740, May 1959.

5. Thieme, H.: "On Fluid-Mechanical Principles for theDetermination of Steering Properties," National ResearchCouncil of Canada, Tech. Translation TT-538 (from Schiff

r and Hafen 9:510-518, 1954).

1. 6. Inoue, S.: "On the Turning of Ships," Memoirs of the

Faculty of Engineering, Kyushu University, Vol. 16, No. 2,j Fukuoka, Japan, 1956.

7. Fedyaevsky, K. K. and Sobolev, G. V.: "Application ofthe Results of Low Aspect-Ratio Win§ Theory to theSolution of Some Steering Problems, Proc. NetherlandsShip Model Basin Symp. on the Behavior of Ships in aSeaway, Wageningen, September 1957.

8. Landweber, L. and Johnson, J. L.: "Prediction of DynamicStability Derivatives of an Elongated Body of Revolution,"

I DTMB Report C-359, May 1951.

9. Lamb, H.: Hydrodynamics, Dover Publications, Sixth! Edition, New York, 1945.

10. Breslin, J. P.: "Derivation of Slender-Body Approxima-tions for Force and Moment Derivatives from Three-Dimensional Singularity Distributions" DL TM 134, January1963.

11. Albring, W.: "Summary Report of Experimental andMathematical Methods for the Determination of Coefficientsof Turning of Bodies of Revolution," Conlan 2.[

R-945

-19-

I

12. Arnstein, K. and Klemperer, W.: "Performance of Air-ships," Aerodynamic Theory, Vol. VI, Durand, W. F.,Editor, Julius Springer, 1934.

13. Flax, A. H. and Lawrence, II. R.: "The Aerodynamics ofLow Aspect-Ratio Wings and Wing Body Combinations,"Cornell Aeronautical Laboratory Report 37, 1951.(Published by the Royal Aeronautical Society, ThirdAnglo-American Aeronautical Conference, September 1951)

14. Gertler, M.: "A Reanalysis of the Original Test Datafor the Taylor Standard Series," DTMB Report 806,March 1954.

15. Bisplinghoff, R. L., Ashley, H. and Halfman, R. L.: Aero-elasticity, Addison-Wesley Publishing Co., Inc.,Cambridge, Massachusetts, 1955, Chapter V.

16. Crabtree, L. F.: "The Formation of Regions of SeparatedFlow on Wing Surfaces," Part II, Royal Aircraft Estab-lishment, Report Aero 2578, July 1957.

17. Prohaska, C. W.: "The Vertical Vibration of Ships,"Shipbuilder and Marine Engine-Builder, October- November1947.

18. Kuerti, G., McFadden, J. A. and Shanks, D.: "VirtualMass of Cylinders with Radial Fins and of PolygonalPrisms," NAVORD Report 2295, January 1952.

19. Davidson, K. S. M. and Schiff, L. I.: "Turning andCourse-Keeping Qualities," Transactions of the Societyof Naval Architects and Marine Engineers, 1946.

20. Eda, H. and Crane, C. L., Jr.: "Research on Ship Con-trollability, Part II: Steering Characteristics of theSeries 60 (Cb = 0.60)" DL Report 923, October 1962.

bIII!j

UR-945 U-20-

H ~ 0 H C, UJLCU\ t-r-CU tr\ cu In CUj 0 ctU -4G\ ny tn)U'\W 0ON41 tl- rA 0 '1 0 0

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I FIGURE 3. MODEL ORIENTATION IN X-Y PLANE

'R9[IiI

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EXPERIMENTAL,

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E 2.92 1959

CALCULATED

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i 12 16 20 24 28 32

I/H

FIGURE 4. 840 SERIES BARE HULLS DERIVED FROM TAYLOR'SI STANDARD SERIES WITHOUT DEADWOOD AFT

PRISMATIC COEFFICIENT Cp = 0.54

I R-945

I _

CALCULATED (1946-47, 1951)

. CALCULATED (1959 DATA)

S~BH : 4.62 1

0i.

0.1 -

I

12 16 20 24 28 32[ 2/H

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[ ~ ~0.1 .}

SYMBOLS WITH FLAGS, 1951

2.92, 1959CALCULATED (1946- 47, 1951)

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[FIGURE 5. 840 SERIES BARE HULLS

R- 945

I-

[ + 0 B/H4.62

1.0 -+ 2.92

.CALCULATEDCALCULATED

0 I I I I I

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a 1. 651SYMBOLS WITH FLAGS, 1951

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IB/H z 4.62

-3#0 - 0 B/H: 2.92 0

+ B/H, 1.85 + _+ S 2.92

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FIGURE 6. 840 SERIES BARE HULLS, STABILITY INDICES[

CALCULATED

CALCULATED[ o., N - N'+ +N'Ii0.1 zN +

I 2.92LCG MOVED .04.2 AFT (1959)

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[ 0.6 I 2.92, 1959

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SFIGURE 7 840 SERIES HULLS WITH FULL SKEG (20)

R9

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; I FIGURE 8. 840 SERIES HULLS WITH FULL SKEG (20)

R-94J5

iI!

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I FIGURE 9. 840 SERIES HULLS WITH FULL SKEG (20)

STABILITY INDICES

R9I

EXPTCALCULAT

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al 0.1

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.PROFILE AREA, SQ. FT.

S~FIGURE t0. COMPARISON OF CALCULATED AND EXPERIMENTALj STABILITY DERIVATIVES AND INDICES FOR 842 HULL WITH

15 VARIOUS SKEGS (199 DATA) S R-7 4o )

R-945

[ 1.0-

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FIGURE 11. COMPARISON OF CALCULATED AND EXPERIMENTALti STABILITY DERIVATIVES AND INDICES FOR 846 HULL WITH-OUT AND WITH SKEG (1951 DATA)

LI R-945

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R-945

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