Simulation analysis of ship motion in waves

Lab. v. Scheepsboijwkunde Technische Hogeschool

Deift _{The Proceedings}

INTERNATIONAL

WORKSHOP ON

SHIP AND PLATFORM

MOTIONS

Edited by Ronald W. Yeung

Sponsored by

The Department of Naval Architecture and Offshore Engineering, University of California, Berkeley

The National Science Foundation The Office of Naval Research

October 26-28, 1983

Continuing Education in Engineering, University Extension, University of California, 2223Fulton Street, Berkeley, CA 94720

ARCHIEF

Contents

Preface

LINEAR THEORIES AND APPLICATIONS Part I

Chairman: H. Chin, Gulf Ol Company Houston, Texas,USA Invited Review: Ship Motionsand Wave Loads, J. P.. Paulling

Strip Methods for Motionsand Wave Loads in Followingand Oblique Seas and Comparison wj.h cxnvriments. S. Takezawa. T. H,rayarna, K.Nishimoto. and K. Kobayaslsi

Wave Induced Mean Shifts in Vertical Absolute andRelative Motions, F. O'Dea

Elastic Response of Shallow-DraftShips Advancing in Short Head Waves. Hosoda, Y. Yamauchs, and K. Taguchi

Discussions Part II

Chainnan. J. V. Wehausen.University of California, Berkeley. USA

Invited Review New 1)irections in Exoerimental Methods,N. Ohkusu and T. Takahashi

Wave Loads on a Floating Solar Pond, T. Miloh

An Analysis of a Wave-PowerAbsorber, G. Nihous and W. C.Webster Approximate Impedance Methods for Wave-Energy Absorptionby Devices in Harbours, D. V. Evans and B. M. Count

Prediction of Bow Flare ImpactPressure by Momentum Slamming Theory J. H. Hwang. Y. J. Kim, K. S. Mm, and S. I.Ahn

Discussions

NONLINEAR SHIP AND PLATFORM MOTIONS Part I

Chainnan: R. W. Van Hoof. Conoco Oil Company. Houston,Texas, USA Invited Review: Second-OrderHydrodynamic Efftcu on Ocean Platforms,

T. F. Ogilvie

205

Drift Forces and Moment on SWATH Ship in Oblique Wavcs,Y. S. Hong 266

Slow Drift Motions byMultiple-Scale Analysis. Y. Agnon and C. C. Mci . 283

Simplifed Three-DimensionalMethod for Calculating Dnft

Forces on Ships

and Scmisubmersiblcs inWaves, P. Kaplan

Nonlinear Ship Motions inShallow Water, R. H. M. Huijsmans and R. P. Dallinga Discussions . 12 - -- 32 49 68 - _. 76 110 132 321 340 Iv 305 Part II

Chairman: C. M. Lee, Office of Naval Research, Washington, D.C., USA On Second-Order Motion and Vertical Drift Forces for Three-Dimensional Bodies in Regular Waves, B. Molin and J. -P. Hairault 344 Hydrodynamic Analysis of Dynamically Positioned Vessels,

H. J. J. van den Boom and U. Nienhuis

Simulation Analysis of Ship Motions in Waves, V. Ankudinov The Effect of Short-Crested Sea on Second-Order Forces and Motion, T. Marrhinsen

Discussions

ASPECTS OF LINEAR AND NONLINEAR HYDRODYNAMICS Part I

Chaimsan: K. F. Beck, University of Michigan, Ann Arbor, USA Invited Renew: Three-Dimensional Wave Interactions with Ships

and Platforms, J. N. Newman 418

An Integral Equation for the Floating Body Problem, T. S. Angell,

G. C. Hsiao, and R. E. Kleinman 443

On the Precision of Various Hydrodynamic Solutions of Two-Dimensional Oscillating Bodies, M. Takagi. H. Furukawu, and K. Takagi 450

Energy Relations in Slender-Ship Theory, P. D. Sclavounos 467 Computations of the Coupled Response of Two Bodies in a Seaway,

J. H. Duncan, R. A. Barr and Y. Z. Lia 491

Part II

Chairman: R. D. Cooper, Flow Research Inc., Silver Spring. Maryland, USA Invited Revicw: Nonlinear Gravity Waves, I-I. C. Yucn 524 On the Problem of Seiche Over Basins of Variable Depth. G. R. Ghanimat.i

and P. M. Naghdi 549

The Shedding of Vorticity by a Circular Cylinder, W.. K. Sob Separated Flow Around Circular Cylinders, B. Pettersen and

0. M. Faltinsen 575 Discussions Lm5t of Participants 602 512 560 599 163 183 202 363 384 404 415

SIPJLATION ANALYSIS c iiP iuricx' IN tVES by Dr. V. Ankudinov' Tracor Hydronautics Maryland, U.S.A. Abstract

The develoçxrnt of a ron-linear mathematical irodel and axputer progrn for predicting the noticn of a ship conducting arbitrary maneuvers in waves is described. The rrcdel includes terms of

both viscc&is and potential nature due to the different nodes of

notion as ll as terns representing interactions between the

ship hull, prcpeller and rudder. So-called urrerrorya effects are included thrc*.igh the use of higher order differential uations with constant coefficients.

ThIs mathematical nodel has been applied to deterministic calcu-lation of seakeeping and ship maneuvering predictions in

irreg-ular seas. The carputer prran is able to predict with reason-able accuracy and ast the behavior of free floating and noored

vessels in a variety of sea and erwironrrEntal conditions, including wind and current.

Introduction

Seakeeping and rraneuverability are both specialized hydrcdynainic

branches dealing with ship notion in the marine envirornt, bet

the zrethods used to predict these notions are quite different.

Theoretical seakeepjng studies which treat wave excitation fran seaway, are usually based on rather sophisticated nathexnatical

techniques, although nost practical results are obtained using

linearized nethcds. Although such analyses have led to remark-able results, there are basic difficulties in using linearized

mthcxs. Camnonly such nethods ignore significant viscous

effects, and the influence of rudders, propellers and

stabiliza-tion and autopilot devices.

In nost cases ship responses to waves are estimated in probabil-istic sense as a function of sea state condition, and real ship notions in deterministic sense are tnikrKn.

On the other hand, in ship maneuvering, the notions of ships in the horizontal plane are governed significantly by the viscosity

of the flow, and it is rot surprising that nost of the

rk in

this

field is semi-empirical. In the standard ship maneuvering

approach it is assurred that the fluid forces are uniquely

deter-mined at any instant and independent of any other details of the

notion except for the gearetrical properties of the ship and physical properties of the fluid. .Therefore the fluid forces can be expanded as Taylor series in xers of the displaceirents,

velocity and accelerations and this leads directly to a set of

linear and ron-linear terms in the equation of notion. It indeed

provides simple mathematical descriptions of the fluid forces, but the force coefficients themselves remain to be determined

either through analytical or experinental technique.

There are fundazTental objections to the asslrrpticn that the

forces are analytical functions of ship gearetry and its simple

eleirents of notion. This weakness of ship maneuvering techniques

is well established, (see, for instance, CuuL&tins, Brard,

Cilvie, Newman, Bishop, et al., Fujino, Frank, et al.,

References 1-7).

It has been shdn that at any instant a fluid

force is determined in part Ly previous notion. So-called

Mnerrory effects", associated with the effects of the free surface

and vorticity, respectively, result in the representation of the

hydxxxlynamic forces arising fran transient ship notions in terms

of a convolution integral over the entire tine history of the

irotion. This situation has been recognized for many years in the field of unsteady aerodynamics and control theory, and recently becane the familiar subject in the marine cnxnunity. Also it has

been established that for standard maneuvers of the conventional

ship in ca]sn water a frequency dependence of the hydrcdyanmic

forces is not very important, and simplified mathematical, experiirental rrodel besed on Taylor series expansion is

justified [6], [7].

For maneuvering at sea the influence of

frequency on

forces is singificant.

The "mrexrory" effects have an

origin fran t

sources: 1. waves effects associated with the

unsteady notion of the hull at the free surface, 2.

vorticity

which is shed fran the oscillating hull. Wave frequency

dependance should rot be confused with Froude number dependancy

which results fran essentially forward notion effects at zero frequency, which is rot oscillatory. But unsteady wave notion do

depend on forward ship speed. Both sources of the "nertory"

-effects are definitely functions of the frequencies. Newman

defined the characteristic nondinensional frequency paraireter associated with the vortex wake as a reduced frequency w t/g while for surface unsteady effects the corresponding pararieter is

w u/g. _{Here w is the radian frequency of the oscillations, L is}

the ship length, u is the forward speed and g is the

gravitational acceleration. For sufficiently snail frequencies

these t

parazreters tend to zero and the pseudo-steady-state

analysis is valid.

Newman's tentative conclusion is that unsteady viscous effects will became significant before

especially for slower vessels. This conclusion is iirportant for analyzing the experimantal results of captive xrt:x3el tests where

both 'Irirory" effects are present; Wave frequencies effects

could be iirportant not only for ship mDtions in irregular sea

waves (where, theoretically speaking, all frequencies are

present) but, also, for instance, for a ship with manual or auto-matic control sensitive to beading and lateral disturbance in

caini water. The viscous frequency effects are beyond

con-sideration in the present paper, and only wave iteirory will be discussed. Also because of practical orientation of the paper,

mathematical analysis will be siirçlified as far as possible.

More details can be fcxird in References 8 and 9.

Tin-Dxriain Solutions with Frequency tpendent Forces

Wa will start with a general nOn-linear description of the rrotion of the çysical - system. If a tirre-dependent input of the system

is specified as X(t), t general response, y(t) can be expanded

into 1xritxenecus functional form as:

y(t) + y1(t) + y2(z) +... + +

1Yo+fh(r).x(t_T)th+

....

.+j...JTh(ri.r2..:.Ta

xx(t - t2)...x(t - r)ddt1...dr. + (4)

This kind of series converges for a bounded input, if the sun of absolute values of all kernel functions is snafler than . Here

0)2,...

i:. .1:

h(r1, r2,... inversely X C

" 1'3

dr2... tha (2)

I N°

h(3. ti,) I I

..

I x "'''dw1 dw2...dca

h(r3,t2... r). is called the generalized impulse response and

H(w1,w21...co) is called the neralized frequency response

functions of n'th order transfer function. In equation (1), x(t) can be a deterministic function, pericdic, transient, or randcxn function.

For the case when n 2, narrely when the system can be assurred as

a second order non-linear system, or quadratic system, for the outpit, patting y0 = 0, and n = 2 in equation (1) can cbtain

y(:)

J

h(t)x(r - r)dr +

J J

h(1.

t)

(4a)

This sun is the expression of linear output, Yl (t) to the input X(t), h(r) being the ixroulse response function

h(t) H(o.,)c" dco ('4a)

arxi quadratic output, y2(t) with h(ti,t2), called quadratic

impulse function.

fl previous discussion was not associated with any hydrodynamic

problem (the system was considered as a TMblack lxxM) and there is

to general methods to carçute the kernels and convolution

integrals (1), (la).

Returning to ship notions problem, we can assume that the iiulse

response function h( t) can be used to cpute the first order

Ixdy notion characteristics (such as displacements, velocities, forces, etc) and non-linear characteristics will be limited to

the quadratic responses h(t1, r2).

The last assniption is quite

logical for potential. fl

where the higher order terme are

associated with velocities squared in Bernoulli's &uation, and

such forces as wave drift forces, prcpuls ion are well described

by the quadratic functionals.

It is also well fitted for the

description of quadratic viscous forces in the first

approxima-tion (cross-coupling transverse viscous forces, separated

flows). Most of the previous studies were basically concentrated on the linear response function. h( r). Quiratic responses

functions are significantly trore difficult to wipute, although

recently, there are ship notions results basically related to the

drift forces, Newman, Ankudinov, Pinkster,' References 10-12. Simplified Method to Evaluate Linear Response Function and

Convolution Integrals

Numerical difficulties to calculate the convolution integrals

Cia) and the integrals with the infinite limits in the frequency

dcznain (2a.) can be avoided using the known prcperties at the

Dirac delta function.

If a function in the frequency &xnain PC w) can be expressed by

the polynanial as

a-i ., 4-L

P(w)C,w cw

w _#...c4

. CIa) Then its Fourier transform in the tine thnain tecares

°° _4-f

where 6(t) is the Dirac delta function. Accordingly the

convoluticn integral can he expressed in the final form siirply as

'i-i 4-I 4-i

jp(tJX(t_Jdri1Z

e(J)

(-i)

die)

The nethcxi can he easily understood fran a sl.Irç)le exaxrçle:

uncoupled hanunic, say .iay rrotion of the Iody, y = y0()e

in the regular wave, with anplitude

ii - _.

il.t"L-"-' Cm _{Aiz(4)#cJ8(w)#c(Q)].}

_-ç4)y)e

_('IL)

where m is the xass of the bDdy,

F2 wf2(w)ei%1t is the wave exciting force

A22(w) is the added mass in sway (a function of the wave

frequency(w), B22(w) is the darrping coefficient and _{C22(w) is the}

hydrostatic term. The inverse Fourier transform of the equation (4a) will give the kncMn convolution intrals

-

JO(r)

(- -r) dt 'JeHg(t-r)dt

*J1Czz(t) y(t-c) dr = _{(.) (Sa)}

which derronstrates the so-called "IreTory" effects in

accelera-tions, velocities and displacexrents. Here

00

J Aj

"i

a (TJZL-_{it 2Z} iJC yft): _{i'-,, ,} "Jr tz/')-

I

8 _('-')e

IJ

_{C (vi.}

k / c.') e

d U "-C (Cq)

Frequency dependent added masses and damping coefficients _{can he} estimated fran the existing seakeeping programs, or fran the test data (P?4 tests, impulse response tests). _{The test results will}

include tx)th waves and viscous "rrexrory" effects. _{For tire-danain}

sinulation _{A22(w), B22(w), C22(w) are expressed Ivj series with} only even power of the frequency w

.

A1(J1xA2

A'J*A('t..

511(w) _L

CzL(') c#D'#.(70)

CD £Z)

where the coefficients _{A22 .}

2' .. C22

_{are constant values.}

Here and everywhere telow the frequency w is the frequency of ship oscillations, i.e., frequency of encounter, or effective

frequency.

The remarkable advantage of the series (7a) is that the cxripli-cated convolution integrals (Sa) will he expressed ty the

ordinary differential equations _{on y(t) higher than second order} with constant coefficients. Indeed

&.

JaftJjt-ca'c. A ITh)j(e-14- AJ.J'j(tC)'iC 1- A&J- i1t-ç)/c_

--at.. it

fl

other terms can he transfornd in a similar manner. Then the

equations (5a) Lecczre a simple linear differential equation of

6th order if the frequency dependent first order hydrodynamic

coefficients are kept up to six order. For nore crnplicated functions one may keep the terms of higher order than six; hcMever in practice there is little need to do it tecause the

derivatives of higher order than sx in ship irtions are very small.

y'.[C,'

A'J4 t(-

8J #[in # A e

j

/jft) (a)

where the excitation functicn F2(t) can include other linear and

nonlinear hull and environrrental forces.

Equation (9a) can he effectively calculated nierically. For

instance substituting

- _{N ()}

pCj

_9L-p

_rzI.

,'o'

it becaTQS i

-,[cW_8p

tc

AJd _{-fin #A_C]_ep C)(4'aJ}

and can he calculated bj ore of the well-knc.in non-linear

nuirer-ical rrethcxls. For a Ix)dy with six degrees of freedan or for a

irultixxiy systen (like t

interconnected ships them) equation

(9a) can be generalized bj introducing the cross-coupling coefficients between different rnxes of notion for eath Lcdy.

Solution can he thtained using matrix inversion techniques.

Equation (9a) has also a simple physical interpretation

considering only the frequency independent terms, cbtain the classiq, equation of notion (in.i. -

Cj- c1i).

The term A2 2(A22 (w = 0) i s the sway added mass of the body in

unlimited fluid (no free surface or waves), arid B2 = B22 (w =

0) is the sway damping coefficient, of viscous origin (wave

dairç)ing in this case being zero). The term C22 (o) is, of course,

C2.

Thus, the remaining terms in equation (9a) represent the effect of the history of notion on present hydrodynainic forces.

According to %hausen [91 the equation of notion for the ixidy in terms of the convolution integral in 6 degrees of noticn can be written as

, coo))9 ji

(ia)

where the added mass in infinite fluid A;KCxu)ard

the retardation

functions kçjtiare related to the added masses and danping in the

frequency dci-nain in the follciing way:

(t) - J4j,( (f) Cj 6t

I(

6. (u) aJ/<'.(it)

It

where BjgJ is the cosine transform of KLt) and (A 1') A (w)iS the

sine transform of KJLt). K,,!i)may be determined either fran (w)

or If one takes the Laplace transform of equations

(9a) and (12a) it is fciind that both equations will give

identical results.

The first-order frequency-dependent wave exciting force on the

right side of the equaticn (7a),

can be expressed ty series similar to that of equation (7a)

ti

z.

-11 '-t 41Yj'W #A tJ#..

EL#L.L (43a)

This force in the tiire dcxrin cars then be expressed as

00

(r)(jt-r,/a't ,Ji'(l)va,

- - -

-

(tii;

C -D ' çtt)- _{(+) +} (EL) . W _{(1) 11)}

where <' the wave amplitude, the wave slope and other

derivatives of the wave amplitude are determined fran the wave

record. Equation (14a) permits the calculation of the wave excitation in a straightforward mariner fran the wave record,

although nurrericafly this will rot be a simple task. For the

siniilations carried out using the present prcraxn, a siirpler

procedure was adopted. The wave spectra was expressed ty the

final form:

& /)

't)=

a (&M) _('W(;c.s( y4, .-zi)# 4;,1.(4k

.1) &c)

where aM,(rJ) is amplitude of k'th wave cxxrçonent with frequency

W, ('-1V) is tJe difference in wave and ship heading angles,

and is a randcnt phase angle. Then each of the K wave

catponents was irultiplied b1' the normalized value of wave

exciting force Fj(t)/a, (j:f.)obtained f ran a frequency xnain

sea-keeping calculations. The value of the wave excitation at any instant is thus simply a sum: %(t)= ±F(LJL)t)

The arputational procedure for sirrulation of ship uotions in

waves is briefly outlined below: 1. Frequency dependent

hydro-dynamic coefficients (added masses, damping, wave ex ting forces etc.) are calculated for enough discrete frequencies to

adequately cover the required range of frequencies corresponding

sea st3es of interest.

2. The constant coefficients A2 ,

... E'are cbtained using series (7a), (12a) and a least

square method. These values are then stored as they will be used

repeatedly for the ship. 3. Tirne-danain sirrnilation are

performed using equations (9a), (lOa), (ha) with a specified

Nutrical SilrLllation of ?tions in the Tirre-flrxnajn

Equations of ttion - The dynamics of a rigid lxxiy can be readily

treated using Newtonian nchanics, bit tI-c dynamics of the

surrounding fluid can be analyzed aily if certain idealizations

are made.

The' equations of notion used for a ship in waves are those of the Standard. SAME Maneuvering Coordinate System, and are fixed in

the ship. This system is carnonly used in maneuvering

sirrnila-tions, bit is different than the translating earth axes normally

used in predicting ship notions in waves. Hc.iever, these body

axis cxx)rdinates yield nuch siirpler expressions for linear hydro-dynamic forces calculated in the standard seakeeping studies. The equations of notion in general form are given in Appendix A. As an exarrle, the equation of notion in sway including the

un.enOryI effect tern is shown in nore detail.

Viscous Lifting Forces - Ships maneuvering in cairn water or in waves experience significant viscous forces and i.ttents which cannot be accounted for by rrthods of potential flow theory.

This effect can be calculated for analytically by using low

aspect ratio wing theory and by regarding the ship hull as a low

aspect ratio lifting s.irface for both the yaw and pitch notions.

Hull lifting forces arise fran the need to satisfy the Kutta

condition at the ships stern.

The principal feature of low

aspect wing ratio theory is that the .side force is calculated by stripwise integration fran the bow to the section with greatest draft (for horizontal notions), or greatest beam (for vertical

notions) as described in Reference 15. The reason why the forces

on that portion of the hull located aft of this point are ignored

can be explained by considering the vortex system. Physically,

this is due to the rapid develoçnt of the boundary layer over

the after part of the ship.

This approximate approach is useful

for calculating the iriportant effect of the viscous flow. In the

-present study the. low aspect ratio results have teen generalized

to harnonically oscillating flow. The sectional forces along the

hull according to strip theory are:

where d,3 and are the sectional added masses and darrpingt"'

respectively, and Lare the section velocities:

Vz 4 _{- U, tr}

-

J

;_ , 6

-These equations are valid for both syimetrical and asynixetrical

contours. Assuming zero value of the local added ness and

danpirrj values at the stern and integrating (lb) over the length,

circulatory forces the terms in Equation (ib) containing the term

u ?j shculd be integrated Iran the bo.z to the section of the greatest span. For irost of conventional ships this section is

very close to the bcw for lateral forces and close to the

mid-ships section for vertical forces.

The total forces acting on

the ship will be the sn of the forces defined ty the Equations

(A-2) and lifting forces defined above.

To denxnstrate hoz the irethod was used, consider the foriiulas for frequency dependent lateral and vertical forces.

& Y;t'.'iJ .1a

Ir

U .J U

d - ALL (t) - Z.

_/

fffl,

('b'(-J

ft f-/.1daa 82Z1/3tjp/

frmci _ze)

L-t4J

fa,d1.fUf"e3

A33f- ui'4,

c 4"ar

(-

2(wi) =

I

dj - q 7

() *

_j

Results of calculations based on Equations (2b) for a Sixty

Series hull of 0.7 block coefficient are presented on Figures 1

to 2.. Sway and heave added masses and damping coefficients for

the ship and their sectional values for largest draft at

and' beam atyz.Sci.. were calculated using the strip theory.

It is

quite remarkable that circulation effects are significant for both borizontal and vertical notions and that the calculated .effect of forward speed confirrrs the trends of rrodel test data.

Linear lifting contrib.itions analagous to those in equations (2b) can be easily found for other inertia and damping coefficients.

The corresponding hydrcdynamic Irarents are calculated ty

integrating the product of sectional forces and longitudinal

coordinate over the leixth. For pitch and yaw, vertical and

lateral velocities

W and V are replaced bj - _and

r.

It can

be easily shciin that for the limiting case wherw-.O and5-'

O or T# 0 _{the Equations in (2b) tecare the Jones' 1CM aspect ratio}

forirulas.

Non-Viscous 1-lydrodynarnic Forces - The accuracy of any

mathernat-ical nodel for ship notions is heavily dependent a-i the

represen-tation of the hydrodynainic forces as a function of body gecntry,

frequency and forward speed.

Perhaps, the reliable test data

will be indispensable for this cxlex problem, and one of the goals of the present paper is an attempt to use the advanced

experiirental techniques (like large amplitude planar irotion

mechanism tests), widely used for standard ship maneuvering

analysis in calm water, for ship sirrulaticn in waves, long time

considered exclusively seakeeping &xnain.

_{As part of this study,}

an extensive analysis of available analytical and exerirrental

results for various

hydrodynamic forces including the

frequency effects has been performed.

Results of this analysis

Experiinta1 data for different ship forms and covering the practical range of frequencies and ship speeds are very limited. The specific tests to cbtain ship and wave induced forces for a

ship with six degrees at freedcrn including the various

cross-coupling and ron-linear effects are needed very nuch for

tine-dcxiain siirulations and further develqxrents of the cczçuter technique in ship riot ion theory.

For relatively low forward speeds ( Pa- the influence of

the Frc*ide number a the added masses, wave dairoing and wave

exciting forces is siall.

Zero speed, three-dinensional czrnputer prograrr are rrost appropriate for this regirre because they

cor-rectly calculate the forces over a wide range of frequencies.

Strip theory calculations can be used for intermediate and high frequencies, bit for low frequencies (especially for heave where the added mass-beccrre infinite)three-dinnsional rrethods or corrections should be used.

Hydrostatic and hydrodynarnic forces should be calculated

using instantaneous positions and not the irean positions of the hull and water surface. Zadeh (16) and Chapman (17) has studied

this problet in sare details.

Similarly, the instantaneous hull

wetted surface should be determined and used a-s the surface of integration for all hydrostatic and hydrodynamic forces. This is not a significant problem for calculation of hydrostatic forces.

However, to express nunerically the hydrodynamic forces as a

function of ship surface is a very difficult task.

Therefore, in

the present study an approximative method, as described below,

was adopted.

For slender ships, the added mass in sway can be expressed

approximately as:

AtW)=

where = sectional coefficient (Saunders, Prohaska)

= Lamb's correctiqn coefficient of accession given approximately by:

g X___ (1- ₎ ( L/5)

T7

accourt for free surface effects. Fran an analysis of Vught's

and Tasax 's calculations and test data it is estimated that

/

a-where a.

z.o.i

the wave damping can likewise he expressed as

821

R4

fr'/

[a(i' - i(±(r) j d

where

4 O.0 4- 0.12. c,.)

For heave the following expressions for added ness can he employed

,

(U'?)

_{./;,/ÔlLiC I)}

L

where the last term corresponds to the added ness in an

infinite

fluid, A (a'). For zero frequençr the abve expression reduces to

A 25 The heave wave damping can he expressed as:

00_(l.T)33

Similar equations were derived for other coefficients and notions. The advantage of these equations is that they expres

the freency dependence arxl the constant coefficients A5 ,..

6j ,..&(see equations (Sa)) as a function of instantaneous heazn

and draft, which can he readily determined at every tine step fran the relative notion hetween ship and waves.

Influence of Forward Speed - For thterirediate forward speeds (F,L) .lc,20) the wave systen due to ship oscillatory and steady not ion beccxne interdependent, and three-dinensional analysis irethods

which include the effect of forward notions, such as that of

Reference 20, probably nust he used. Nurerical analysis of Q4 data with horizontal plane notions (obtained at low frequencies) show, that the increase of sway and yaw forces with increasing forward speed is basically due to increased sinkage and trim.

'igures 3 and 4 show the data cbtained at Thacor Hydronautics, for a containership, at 12 and 23 knots and estimated data for 23 knots conditions cbtained using the 12 knots data with

correc-tions for sinkage (new draft) arid trim (quite snail for this

example). Trim corrections were those reported in Reference 15.

fl sway and yaw damping forces were expressed as:

Y&)a

if['c

_{(4#L(f)'f,))v]#[}

_{.r"iVi. cjj}

jWv)' fi LZUII14VI K, (4-i () 31J)jv'J [4jvWi &i/

Y(r) .p £ 'i//I 'K (4. _.

N(rJ LtC/t( (i ().(_f)r'J [iY/rhlI-h/.kj)

where the standard maneuvering definitions are used, and 0 is the static trim (in radians). Inertia forces were corrected in

proportion to the draft squared. Analysis of test data for

swaying and yawing of 0.70 block coefficient Series 60 hulls with I/B of 4, 5.5, 7, 10, and 20 at Frc*icie numbers of 0.15, 0.20 and

0.30 (See Reference 19) shcwed trends which were similar to those

for the containership. These results confirm the inportance of calculating hydrodynainic forces in tilTe dcxnain, using

instanta-neous ship Ixsition and wetted surface.

Estimation of Wave Induced Forces - First order wave exciting forces and nm2nts were calculated using a strip wise technique

and the TMrelative' speed between the hull and waves It was assumed that the wave-induced force on a ship could be ccrriputed in the saire way as the notion-induced hydrodynamic force if one

defines the effective relative velocity between the hull and

water surface as tie - u, ,

-where the wave velocities iJ,,, V, include the nith effect.

Cczrpariscn of measured wave forces and wave forces calculated

using this concept ty Gerritra (14) and Vughts (13) s1w gciod

agreémant for both vertical and horizontal irotions. Wave

exciting forces were calculated fran the general expressions:

'' I/(a'.-!_

,W(')a F,j

#j

z., ui' Lj

1 ₄₀

It) c',("A) F 4k,) c

(Jt r I,)

flz4'

jadratic wave drift forces, including slcily varying drift force carponents were calculated using results of the previous studies

conducted at Thacor Hydronautics. Non-linear cross flcw and cross-coupling coefficients were assumad to be frequency

independent, and the velocities associated with these tern were approximated using the re1ativeM speed at the center of

gravity. Wave exciting forces are corrected not only for the nith effect bit also for the effect of finite length using KIL

(for surge, sway, heave, roll ter) andb(for pitch and yaw). The factors k and K.L are taken fran Haind (21). se can be represented approximately bj

*',.z e

cost.)

,

K: e COS(o.'i ),. '

(a

= waterline coefficient )

A nore accurate ccxnputaticn of these forces can probably be made

by sinply integrating these ron-linear forces over the hull

length: /

ho,--where all velocities are relative velocities taken at the

speci-fied rosition J. The value of the coefficients and

Results of Nmerical Simulations

The present tirrQ-danain notion simulation program is simplified

tq uncciipling the vertical notions (heave and pitch) fran the

surge-sway-roll--yaw rrotions. A critplete six degree of freedan

notions sinulaticn is not only requiçes much greater caiputer

time, but perhaps nore important, reqtiires a knowledge of oss-ccxipling coefficients which are not normally neasured in ship maneuvering tests and are alirost impossible to predict

theoreti-cally. Meaningful experirrental efforts in this area are urgently

needed. Results of tirre-dcxnain calculations were xzipared with

calculations based on standard strip theory (such as SMP 81,

Reference (20). Results of the cxxnparison for pitch notion in

irregular long-crested head waves are shown in Figure 5

Results fran the present simulations, f ran frequency-domain sea-keeping program., SMP and fran tests, conducted at Thacor

Hydronautics, are quite similar. The differences between the

present sirrulaticn results and SMP81 results are perhaps due to the different pitch damping, which in the present calculations have been corrected for three dirrensional effect and also

includes lifting effects, rot included in the standard seakeeping

programs.

Maneuvering simulations in ca]m water and in wave were carried

cut for a Mariner. Extensive low frequency, calm water F4 tests, of this ship were conducted at Tracor Hydronautics, Reference (19) and other laboratories, and resulting simulated maneuvers were quite close to trial data. 1n addition, a large

exper±tental program for this ship was conducted by Frank,

Loeser, et al, [7] using the impulse test technique. The

hydro-dynamic coefficients for low frequencies (w-o) were taken fran (19), and the appropriate frequency related constant coefficients

A1S(ref1ected in the higher order derivatives) were estimated fran the strip theory. In the present simulations no corrections for instantaneous bydrodynamic forces were made. 1. prcpeller

and rudder forces were expressed by:

Xpu,p

I

pLL[Q L 'CziJ = 4jL[d u- e; un

were the coefficients a J which are defined for various u and

propeller rotation speeds n, were corrected for wave velocity at

:--L through the effective velocity U

The limited scope of this paper cbes rot permit discussion of

other features of the carputations. The ship notions program

itself is the integral part of large simulation system developed

in Tracor Hydronautics. This simulation can nodel alirost any environirental disturbances. The cxxrputer program is run on a CEX

VAX-750 ccirçuter system, and for typical tiire step ATzD.Dr5e. a

typical ratio of the ccnputer simulation tiire to the real tiire is

about 1:3. Thus, to obtain typical ship notion record of 30 minutes duration will require about 10 minutes of (J tima. Ship trajectories in turn and 20 x 20 zig-zag maneuvers in calm water and in irregular long-crested waves (wind forces were set at

zero) for the Mariner till are given in Figures Ga and C.

For relatively small waves (significant wave heights

tlq,c5ffl the cniy distortion in the ship trajectories is a

general drift. Printcut of details of ship notions shci that

even for small wave teights significant and rapid tiire varia-tions of yaw rate and drift angles occur. For larger waves

heights the ship behaves quite erratically and at certain

situations can lose stability, as denonstrated by the turn

for H £1Lf and a zig-zag maneuver for H,6

$f

Sumary and Recarnendat ions

To date there have teen only limited efforts such as

References 22 and 23 to deterministically describe ship

notions at sea. The rrost widely used approach has teen the

use of convolution integral. As an alternative to carputing these difficult convolution integrals, which require

kncMledge of the Kernel function for all frequencies fran

zero to infinity, a practical sirruilation can be carried cut

cnly for the frequency range of practical interest. This can

be thne using ordinary differential equations with constant

coefficients of order six or less.

The required coefficients

can be determined by curve fitting fran existing test or

calculated data. Tte required caiputer tirre is only nodestly

greater than that required for standard ship maneuvering

sinulations with constant coefficients and is quite reason-able for practical engineering requirerrents.

The present rrethod and czxnputer programs cxrrines airrost all standard frequency-dcxnain seakeeping effects with additional

terms accounting for linear lift and ron-linear viscous oross

flcw forces as well as azrplitude effects, performance of rudder/propeller and so on. As a result of inclusion of these effects the present methcx is expected to be nore

accurate than existing seakeeping programs.

Hever, to

fully realize the potential of this method, significant

additional experimental and analytical efforts, should he

undertaken.

References

1 Oxrinins, W. E., "The Impulse Response Function and Ship

Motions," Schiffstechnik, Vol. 9, 1962,

Brard, R., NA Vortex Theory for the Maneuvering Ship With Respect to the History of Her Motion,w 15 Symp. on Naval

Hydrodynamics, 1964.

Oi1vie, T. F., "Recent Progress TcMard the Understanding

and Prediction of Ship Motions,TM Proceedings, 5th

Syirposiun Co Naval Hydrodynamics, 1964.

4 Newman, J. N., "Sare Hydrodynamic Aspects of Ship

Bishop, R. E., Burcher, R. K., and Price, W. G., "The Uses of Functional Analysis in Ship Dynamics," Proc. R. Soc., A, 332, 1973.

Fujino, M., "The Effect of Frequency Ependence of the Stability rivatives on Maneuvering Motions," ISP,

Vol. 22, 1975.

Frank, T., Lceser, D. J., Scragg, C. A., Sibul, 0. 1., Webster, W. C., and Wehausen, J. V.,

"TransientManeuver Testing and the Equations of

Maneuvering," Proceedings, 11th Syi:posiun on Naval Hydrodynamics, London, 1976.

Tick, L., "Differential Equations with Frequency Ipendent Coefficients, JSR, Vol. 3, 2, 1959.

Wehausen, J. V., "The Motion of Floating Bodies," Annual Review of Fluid Mechanics, 1971.

Newman, J. H., "The Drift Force and Mcinent on Ships in Waves," JSR, Vol. 11, No. 1, 1967.

Ankudinov, V. K., "None-Periodical Forces and Mczrnts on Ship in Waves, ISP, 76, 1969.

Pinkster, T. A., "Mean and Lcw Frequency Wave: Drifting

Forces on Floating Structure," Ocean Engineering, Vol.

6, 1979.

Vughts, J. "The Hydrodynainic Forces and Ship Motions in Waves, Dissertation, Deift, 1970.

Gerrita, J., et al., "Effect of Beam on the

Hydrodynraic Characteristics of Ship Hulls," 10th Syrnp. on. Naval Hydrodynamics, MIT, 1974.

Fedjaevsky, K. K. and Sobulev, G. V., "Control and Stability on Ship Design," Translation fran Ressia,

1964.

Zadeh, "Linear System Theory," McGraw Hill, NY, 1963. thaçmn, R. B., "Large Anplitude Transient MotIcn of Two-Diitnsional Floating Bodies," JSR, Vol 23, No. 1,

79.

Meyers, W., )pp1ebee, T., and Baitis, A., "User's Manual for Standard Ship Motion Program, SMP"

E11'NSRLC/SPD-0936-0l Rep., 1981.

Goodman, A., et al., "Experimental Techniques and Methods of Analysis Used at Hydronautics for Surface Maneuvering Predictions," HYDRtAW'ICS, Incorporated Report 17600-1, 1976.

Chang, M. S., "Canputations of Three-Dimensional Ship Motions with Forward Speed," 24 mt.

Syp. on Nn

Hydrodynamics, 1977.

Haskind M. D., "Ship Hydrodynamics," in &zssian, 1973. Perez, Y. Perez, L., "A Time Solution in the Motion of Steered Ship in Waves," JSR, Vol. 18, No. 1, 1974. Bc,urianoff, C., "Caiutational Prediction of Ship Motion

in Continued Waters Using Inertial Marker Particles, MARAD Report 77084A, 1978.

Schmitke, R., "Ship Sway, Roll and Yaw Motion in Oblique Seas, TSNAME, Vol. 86, 1978.

Appendix A - Transformation of the Coordinate Syst and

Equations of ktions

Cartesian coordinates (X01Y0120) are taken to he fixed

in space with z0

is vertically biriward.

The associated

coordinates fixed with respect to the ship are denoted (x,y,z), the right-hand convention is applied so that y is

positive to startoard.

The linear velocity (u,v,w), and

angular velocity(p,q,r) are defined in relation to the ship

coordinates (x,y,z). ?rgular orientation of the ship is defined by the symbols ₀ (roll), 9 (pitch) and (yaw), in accordance with the follcMing coordinate transformation:

xcos8 cos siltIcOiyf - coØ silt1) L(cosO aOw7V-sia/CD.Sy)

XcoS9 ylcos c.sy r.infJia.!fr J4.OJ o.9..u4O c'ssi41u4y)

2 -ysino y cosO s; +i-c45D

c4f

To the first order of the potential hydrodynamic forces, the

equation of notions in the bcxiy axis coordinate systn (in vector rx)tations) are _{-. -,}

-, -,

-( m_

4x(7)Fft)

___jl,t,___

(Im)*UX()

U) (A-I) Here Km and are the translational and angular body

3uXntum, respectively.

'(8, 83,83) - ffcp7e d$ , 3 l8y Sc, B) -fJq'(PV)d$

are the vectors of the instantarous pressure ixrpulses over

the ship surface.

The vectors Fit) and1t) include the rest

of the ship forces (hydrostatic, viscous, non-linear terms of

different nature, rudder, propellers, envirorurent, etc.).

For a ship noving with irean speed U0 and having snail

harironic irotions in waves, angular velocities and

accelerations are identical for the body axis and earth coordinates txit translational velocities and accelerations are different:

L4V.U.Y

where

_{f and f}

are sway and heave displaôennt in translating

earth axes. 'Introducing the force terms

-

cc:

Jf

?Yds

U,[A;-we obtain the follciing equations of notion (except surge) in

the two coordinate systems:

Stability Axes Translating Earth (Seakeeping) Axes m)i r8,Vt

A2t 82

fj

(A2# rA4J #8,

I. '. W_t53W 8 m.1,)9

*fi (A+I)

'

t -z

?'

(Acs-+rr)ë t

I( Asjm)

LI. '33)O 4 _. F.' + _jf #8 j' .s

#(63u. A)

W _{11 A,39 .(15,r} '1.A,3)j, q.51

,

N. _u.&.y

.(8J

.s A1.U(g

,4i

(A-2.)

_.IA-;)

11 _{normal seakeepirr equations (A-3) are well known,}

(131,

(14). _{The equations (A-2) for the lateral rroticns are}

identical to those obtained by Schrnitke _{(24]. The frequency} dependent terms in the equations _{n be expressed by series}

in frequency w for tiire dcmain _{sinulations, as in equation}

(7a).

Sway &uation (in the fixed in ship coordinate system)

+ W + XGfJ

f

+ + L'(Y'vr)

1 L2LY,'uv +

!'v1vj vfvl +L T 'wJ(-1)xI

+L'(Y,'ia+Y'uv+Tr 'vf'4)

+1L(Y'_{2 r,ga}

''tJ

+L'(T;,r

Vç(çL/ujJ1+ç, 'viJ

V1)c1 ₍₎₇

+ Yggp t

Yprpp +Yw;r,d+ Y.d-;j- +

(I'ay

W# ?nem.e/ J

L3( (j)7.lL'n'11'.a A1' (1)lIl

It

JL1(As(T)2uL1[y1(jJ&(r)v

_VL[J4/3d)r)

£

-LT(ZfrJ4f±L3Ta&f)it 1&)1_.

4 _{J, UL'(Y'(.JJ o'[f,)}

tUh['(I)J GJ(1

1

Terms _{are associated with added masses and} dazrçing in the frequency &main as shown below: 1w : w

(- ),'(o)J(4-A2' kJ'.A _{w'Y3-fUT/?[(ø_} (j)"z2

U LL

_Y(W):L7'ji

- Yr L[yi(0)3f 1_$w

A))e4 ULTq?('

-..y.

fLfT[6w.. &c'.'j f

uL-}'(D)2j1

flci!i

Constant terms A22 ,..B26 are for the whole ship

Constant terms a221.b2 are for the section at 0.05 1.

Terms Y.J-Y!,Yo) Ye', y/(o).

',

.'(o)r

!

are standard ship

1 -16 V V -120 -80, kgf sec m -40, -0.016 -0. 012

"I

_I.

V -0.008' -0. 004. TLST DATA (VUCTS) o 0 F,. 0 VUCTS. V o 0 F 0.2 TEST DATA A A 0.3 -- PRESENT CALCULATION

FIGURE 2 - COMPARISON OF CALCULCATED AND EXPERIMENTAL LATERAL ADDED MASS COEFFICIENTS Y.'. MODEL 60

SERIES, CB = 0.70. V 10 y.

I 10

I.12ffi I. ID .S.S3i. P .1.21_a

F.LI CLOU FIT

sTMIP THtO*Y ISUS FOAM

t - Ii FEET. REFERENCE .ai

--13

'.-/ ..-.

4/'

/

--S

II

I-

-S -S -S

1-/

.5- .5---P*S(NT Pa CALCULATION

/

0 2 4 6

.8

10 12

CIRCULAR FREQUENCY, sec

FIGURE 1 - COMPARISON OF THE CALCULATED AND

EXPERIMENTAL VALUES OF V MODEL

60 SERIES, CB 0.70. V

0 1 3 4

wV[7

>. 0.6

z

w U U.. U-Ui o 0.4 C) Ui U

0

U-_ 0.2 Ui 12 KNOTS - TEST OATA

----&-- TRACOR HYDRONAUTICS ---ESTIMATED FROM 12 KNOTS

DATA WITH CORRECTION ON SINKACE AND TRIM

//

/

/

Y. - .

,

/

, N

1/ £''

A

/

-,.

,

-/ K

1 1 I i I I I I I

,

,

0. 3 0.2 0. 1

FIGURE 4 - VARIATION OF YAWING MOMENT COEFFICIENT WITH

r' FOR CONTAINERSHIP OPERATING AT 12 AND

23 KNOTS FIGURE 3

z

VARIATION 0. 4, 1 0 COEFFICIENTS SHIP OPERATING

OF LATERAL FORCE AND YAWING MOMENT

WITH DRIFT ANGLE FOR CONTAINER AT 12 AND 23 KNOTS

2

23 KNOTS U KNOTS

A----

_{ESTIMATED FROM TEST DATA}

AT 12 KNOTS WITH

CORRECT-

I-z

0.2

Ui TIONS ON SINKACE AND TRIM

U _{0 I I} Li. IL. Ui

0

U

0.2

I-.

z

Ui

0.4

0

0.6

0 z _0.8o _{0.2 0.4 0.6 0.8 hO 1.2} YAWiNG ANGULAR VELOCITY, r' rL/u

0 8x 1 0 2 Y,N'

23 KNOTS

á-4.

0 4 8 12 16 20 24

25' LU ₂₀

z

Ui 10

z

ow

U a 0 NO WAVES - VAVES5FT

SOLID LINES ARE SUP-si PREDICTIONS DOTTED LINES ARE PRESENT TIME DOMAIN CALCULATION SS TO - ii SEC SSS T05s.c

-.

-,

A A f I I I I I 5 10 15 20 25 30 35 140 SHIP SPEED, KTS

FIGURE 5 - PITCHING MOTION OF FINE HULL SHIP (C _{= 0.50)}

IN LONG-CRESTED SEAS B

-.. WAVE DIRECTION

WAVES B

b. 35 DEGREE RUDDER TURN

(CURE 6 - SIMULATED MANEUVERS OF MARINER IN CALM WATER AND IN IRREGULAR WAVES FOR 15 KNOTS APPROACH SPEED SS. TO - 7 s.c NO WAVES ZIC-ZAC ZIC-ZAC IN WAVES SFT ZJC-ZAC IN WAVES 8FT a. 20-20 ZIG-ZAG MANEUVER HEAD REACH WAVES UFT MODAL PERIOD

TO SEA STATE TEST

1.7 sec

11.1 s.c S 0

000