Two-dimensional damping coefficients from thin-ship theory

ARCHIEF

TWO-DIMENSIONAL DAMPING COEFFICIENTS FROM THIN-SHIP THEORY

by

Paul. Kaplan

and

Winnjfred R. ¿acobs j:

STEVENS INSTITUTE OF TECHNOLOGY DAVIDSON LABORATORY

CASTLE POINT STAT1ON Lab.

y.

Scheepsbouwkunde

HOBOKEN, NEW JERSEY

Technische Hogeschool

TWO-DIMENSIONAL DAING EFFICIENTS

FROM THIN-SHIP THEORY1

by

2 Paul. Kaplan

and

Winnifred R. Jacobs3

This work was carried out at the Davidson Laboratory of Stevens Institute of Technology under support of the Society of Naval Architects and Marine Engineers, Panel 13-5 on Analytical Ship

Wave Relations.

Chief HydrodynamiciSt9 TRG, Inc. (Technical Research Group),

Syosset, N. Y. (formerly Head, Fluid Dynamics Division, Davidson Laboratory, Stevens Institute of Technology, Hoboken, N. J.)

Mathematician, Davidson Laboratory, Stevens Institute of Technology,

ABSTRACI

Analytical expressions are presented for the two dimensional section damping of ships, which are based upon thin-ship theory. The results are applicable to forms with sloping

sides

at the waterline, thereby complementing

other available damping data. Application of these results to realistic (and full) ship forms shows fair agreement with experiment and with the results of other theoretical computations0

NOMENCLATURE

dimensionless wave amplitude

a wave amplitude at infinity

B beam

c _{wave propagation speed}

f ship section lateral offset G Green's function

g gravity

2

K wave number

og

N total heave damping coefficient

N9 total pitch damping coefficient

n _{normal; also general numerica], exponent}

n _{sectional damping coefficient}

R radius of circle T draft of section

t time

w vertical velocity amplitude

x longitudinal coordinate along ship y lateral coordinate

z _{vertical coordinate}

z amplitude of vertical motion

p water density

'0 velocity potential circular frequency

INTRODUCTION

When considering the steady state response of a dynamic system to an oscillatory

forcing function, it is

evident that

the

damping coefficient is the quantity that

limits the response at resonance conditions.

This situation,

with some minor modifications due to cross-coupling, is

definitely true for the case of the motion of a ship in a

seaway. The importance of this

quantity, and also the fact

that it is amenable to mathematical solution by use of

potential flow

theory, has led many investigators to

attempt

a theoretical solution. At the present time, the moat

successful method of calculating the

damping for a particular

ship form is by use of strip theory. In that method, the local damping at a section is found from the radiated wave energy, assuming that the body and the flow field are two-dimensional. An integration of these strip values

over the ship length gives

the total damping. There is no interaction between different

sections and also no account is taken of

forward speed.

In the practical application of the strip method of cpmputing damping, a value of sectional damping is obtained from a family of curves for different forms by interpolation. The results of Grim l4for Lewis sections have been found to be the most useful set of data,

where the sectional damping

coefficient is plotted in terms of the beam-draft ratio and the sectional area coefficient.. The Lewis sections, obtained

from Lewis' (2) basic transformation of a circle, are all

wal]..»sided at their intersection with the surface.

Thus, any

important effects due to the nature of sloping sides at the intersection with the free surface cannot be found from the data for the Lewis family of sections.

The methods used by Grim to obtain values of the sectional damping involved extensive numerical work in nen

analytic form for each section. It would certainly be

advantageous to have a simple analytic forimila which could

be used to obtain the damping for any arbitrary section. In a discussion of a paper by Havelock (3)), Vessera (4) gave an

expression r the two-dimensional section damping of a thin

ship. The final result, was presented in simple analytic

form, but no details of the procedure (use of Fourier transform

methods) were. given.

In the present paper it is intended to derive the result for section damping of thin ships by another method, to present charts for a fmily of various representative sectional forms, and to apply the results to practical ships in order to determine the range of applicability of the results.

SOLUTION OF THE POTENTIAL PROBLEM

-The method of solution of the potential problem is by application of the Green's function technique. _{The basic}

equation for the potential is _{aplace's equation and the}

boundary conditions for the potential and the Green!s function

uRÇ,

A-E

_y

cì

G(iì,C; y, z) are:

A7

)

o

(2)

where the coordinate axes and half of the thin ship section

are shown in the sketch below.

The quantity K0 is defined

as

2

where w is the circular frequency of oscillation, w is the

vertical velocity amplitude and f(z;x)

is the lateral offset

of the section at some particular station (x-coordinate

location). The potential is represented as

Ø(y,z)et

Let L

denote the positive real axis,

L1 the

positive imaginary axis (directed downward), and A the

projection of the sectional contour on the imaginary axis.

Only

half of the ship section is considered due to

the

symmetry of the problem.

Applying Green's theorem

q5 ₁

)&

2Ls

₍₇₎

around. the contour represented by L1, the quadrant of a large

semicircle, and.L2, it can be shown that the contributions

from the circular arc portion vanish because of the asymptotic

behavior of the potential and the Green's function.

The

contributions along

L2 cancel identically, and the resulting

value of

Ø

is

= (8)

where

the boundary condition on the body surface (applied

en the projection on the

axis) has been used,

and T

is the

draft.

The Green's function

2-(-

í-fl

4S

d

(9)

G=4I

₍

L-r

1-(Y-

_o

4-(4

satisfies the free surface boundary condition, but not the velocity boundary condition (equation (4)). Another Green's

function satisfying

the free surface boundary condition is

and the sum of the two is the required Green's function to be used in equation (8), i.e.

()

G1

+-The part of the potential that leads to the radiated waves at infinity is the only quantity of interest in the present

Since the wave height ii

(10)

of the

(13)

and the K'integra]. in equation (12) for large values of ja

iK0y

the amplitude of the waves generated to both sides body is, letting w

iciz0,

=

study, and it is represented by

jL4Jt i

and the sectional damping coefficient is then

T

f

-which is in

agreement with Vessare' result.

The total heave

damping

coefficient for

a ship, according to strip theory, is then represented by

N%

and the total pitch damping by

=

L

where xband x are the bow and stern x-coordinates,

respectively.

EVALUATION OF DAMPING FOR A FAMILY OF SECTIONAL FORMS

In the application of theoretical damping data to ship motion problems, the damping coefficient of a segment is represented by

-i

h (19) where A & (20) (16)

-6-The mean energy radiated to both sides of the body

is therefore

is the ratio

of the

radiated

wave amplitude

to the amplitude of the heaving motion of the ship section. The

quantity A

is presented in mast of the charts auarizing theoretical results, as in the work of Grim (1], and that will be the form in which the results of the present study are presented.

The family of sections chosen was the set of forma considered by Haskind

(NJ, in which the sections

are

repre-sented by f = Z(), where B is the beam at the particular section, and where

\

Z(=

(21)

The values of n considered were n - , l, ., 2,

3_and 4 and

the beam-draft ratios were taken from 0.5 to_2.0, in

steps of 0.5. The value of A2 may be represted as

T

2..

I

-

(i3)

[f

-'

--r

e

(22)

As an example, the value of the integral

in the expression

for A2 will be evaluated in detail below.

T

r

o

Letting K0z y2 the integral becomes

ÇL

Í1

-where

is the error function

connected with probability

theory. The values of A, according to the definition given in equation (22),

are exprea8ed in analytic

form

in Table i foriáll values of n

considered.

Curves

of

A as a function of 1(B/2

2ß/2g for

different values of n and beam-draft ratios are

presented. in

Figures 1 through 4, where the ship forms for each case are

aleo drawn.

The

quantity is the

gecdn

coefficient, given by

n/ui-l.

Thus a new

tool is available for use in determining section damping of forms different from these

considered in

previous

studies. It only remains to

determine

the range of

applicability, of these results to realistic forms, in view of

the fact that they were derived on the basis of thin ship

theory assumptions.

DISCUSSION AND CONCLUSIONS

As a test

to

determine thá range of forms to which these results could be applied, it was decided to consider the results obtained for a section which was a semicircle. A forni

of this

type is quite full

and

certainly violates the

oenditions

required for thin ship theory tóbe valid.

However,

theoretical results for this case are available in the werks of Grim and Ursell (6] and comparison would give a good measure of hew far

the thin ship results would differ from

more

exact theories.

Fer a circle of radius R, the value of

A is

and this was evaluated

numericálly in the

range 0K0R

2.0,

whore K0R w2B/2g

for

the present case. Figure 5 presents a

comparison of these results with those of

Grim and

Ursell,

and

while

it is obvious that the

present results differ in _magnii..

tude from. the others, it is seen that the manner of variation

is similar while the magnitude departure is not so severe as

'would be anticipated for such a form.

This it appears that

the present results can be used as a tool in order to find

dampinglues for representative ship forms.

The charts in Figures 1 to 4 were used to obtain

values by interpolation for the section damping of the Series

O model tt aa studie4 experimentally by Gerritsma

t

7]i.

M integration ever the. hullis used t. obtain, the total heave

and pitch damping, and the results are shown in Figure 6,

where comparisons are

made with experimental data, the value

from Grim's charts,

and

also the value obtained from a

circum-ferential source distribution, Iown as the HavelockHelstein

method (8).

It may be seen that the thin ship results are in

better agreement with experiment fer heave damping than those

derived from the use of Grim's charts, while the pitch damping

La larger than that obtained from experiments and Grim's theory.

Thus it appears that the results derived from .thïn ship theory

can be utilized with fair Success to determine heave and pitch

damping for ueual ship forma, as a complement to the Grim

charts, while their best application will be te

forms having

Sloping aides at the waterline interSection. _{The ability to}

determine sectional damping values analytically by this method

lends it

some attractiveness for future work.

This present investigation was part of an overall study program whose purpose was to determine the

three-dimensional stripwise

corrections for the distribution of

damping along a

ship hull. These results are necessary in

order to achieve mere precise computations of bending moments acting on ships in waves, and also for a better understanding

of damping phenomena. The two-dimensional strip damping values found fer thin ships aré therefore available to be compared with the

three-dimensional

strip values, which account for

in-teraction between different parts of the hull. (The results

for three-dimensional strip damping values will be presented

in a forthcoming note). From the aforementioned comparison, the major effects of three-dimensional flew

at every

section of

an oscillating ship (within the limits of thin ship theory) will be delineated. Thus a better knowledge of the actual

damping force distribution will be Obtained, and this will assist in carrying out bending moment and

motion

calculations in waves with greater accuracy.

-REFERENCES

Grim, O., "Berechnung der Durch Schwingungen Eines Schiffskrpers

Erzeugten Hydrodynamischen Kräfte", Jahrbuch S.T.G., Vo].. 47, 1953. Lewis, F.M.: "The Inertia of the Water Surrounding a Vibrating

Ship", Trans. SNAME, Vol. 37, 1929.

1-lavelock, T.H.: "The Damping of Heave and Pitch: A Comparison of

Two-Dimensional and Three-Dimensional Calculations", Trans. of Inst. of Naval Arch., Vol. 98, No. 4, October 1956.

Vossers, G.: Discussion of Have].ock's 1956 paper. Trans. of Inst.

of Naval Arch., Vol. 98, No. 4, October 1956.

Haskind, M.D.: "Two Papers on the Hydrodynamic Theory of Heaving and Pitching of a Ship", SNAME Technical and Research Bulletin

No. 1-12, April 1953.

Ursell, F.: "On the Heaving Motions of a Circular Cylinder on the Surface of a Fluid", Quart. Jour. of Mech. and Applied Math.,

Vol. 2, June 1949.

Gerritsma, J.: "Experimental Determination of Damping, Added Mass and Added Mass Moment of Inertia of a Ship Model", International

Shipbuilding Progress, 1957.

St. Denis, M.: "On Sustained Sea Speed", Trans. SNAME, Vol. 59, 1951.

3 4 TABLE i -K T 1

[2 - e

(K 2T2+2K T + 2))

(K0B)2 o o -K T

[6-e

°

_{(K3T3+3K2T2+6KT+6))}

o o o

-

12-1/2 1 2 ' B 1/2

[erf(\ji)]

o

e°)

()(1-o

References

Kaplan, Paul: "Application of Slend3r-Body Theory to the Forces Acting on Submerged Bodies and Surface Ships in'Reular Waves", Jour, of

Ship Research, Vol. 1, No. 3, November 1957.

Kaplan, Paul, and Hu, Pung Nien: "The Forces Acting on Slender Submerged

Bodies and Body-Appendage Combinations in Oblique Waves't, Proc.

Third U. S. National Corç. Applied Mechanics, 1958.

Hvelock,. T. H.: "TheForces on a Submerged Body Moving Under Waves", Trans. Inst. of Naval Architects, Vôl.. 96, 1954.

Cummins, William E.: "Hydrodynamic Förces. and Moments Acting on a Slender Body of Revolution Moving Under a Regular Train of Waves",

David Taylor Model Basin Report 9l0,.December 1954.

5.. Lighthill, M. J.: "Methods for Predicting Phenomena in the High Speed Flow of Gases", Jour. Aero. Sci. VOl. 16, pp. 64-83, 1949.

6.', Spreiter, John R.: "On S1nder Wing-Body Theory", Jour. Aero. Sci.,

Vol. 19, pp. 571-572, 1952.

Cummins., William E.: "The Force and Moment on a Body in a Time-Varying Potential Flow", Jour, of Ship Research, Vol. 1., No. 1, April 1957.

Kuerti, G.; McFadden, J.A. and Shanks, D.: "Virtual Mass of Cylinders

with Radial Fins and of Polygonal Prisms", U. S. Naval Ordnance

Lab.', NAVORD Rpt. 2295, January 29, 1952.

Taylor, G:. I.: "The. Forces on a Body Placed in a Curved or Converging

Stream of Fluid",Proc. Royal Soc., A, London, Vol. 120, 1928. 10. Landweber, L. and Yih, C. S.: "Forces., Moments, and Added Masses for

Rankjne Bodies", Jour, of Fluid Mech., Vol. 1, Pt. 3, September

1956. . .

N-543 20