Relative bow motion and frequency of slamming of SWATH cross-structure

DAVID W TAYLOR NAVAL SHIP

RESEARCH AND DEVELOPMENT CENTER

TECHNISCHE UNIVERSI Laboratorium voor Scheepshydromechanica Archief Mekelweg 2, 2628 CD Deift Tel.: 015-786873-Fax: 015-781838

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A study was

xade of etb&4s

for iprof ng the nteans of

estinatirtg

the expected ctther

of wa-ge i=pacts per

unit tine of

the SWAT3 ship cross-s trtxcture. To avenrnes

were explored:

(t) inprovennt

of relative

notion esUnates by adding the

con-poe-tirs of ship-generated wave atid diffracted wave to the

md-- dent wave in describing the free

surface:

and, (2) inclndirtg a limiting impact angle in the criteria defining the occurrence of a saim in the fornalation of the level-crossing definition from which the expected ntuaber of impacts

is derived.

_The

_{results of}

this study sfii that including the ship-generated we and

diffracted wave does not improve the correlation of the ccmpcted relative notion with results obtained from

experiments.

Imposing a limiting impact

angle on the

definitfon of cross-strtture

slamming, as expected, reduces the estimated frequency of slaiiming.

Mditiocal

model exerLxtents

are recommended to obtain a more

definitive estimate of threshold velocity and limiting impact angle for estimating SWATH cross-structure sIamrdtrg frecencies of

occurrence

-A1NTSTAflVE IGtLTIO

This work was performed under the aval Sea Systems Cotmirand General

Hydrody-namie

!asearch

Program a ministered

by the DTSDC Ship Performance

Department. Funding was provided under Program Element 611 533g, Task Area SP23O1!O1, and Work Unit 1562-500.

INWc3CTIO

A potentially serious problem inherent in the SWATH ship concept is its propensity for sustaining wave impacts on the underside of the structure con-nectirig the twin hulls while operating in * heavy seaway. The Impacts not only can Impose large tertiary loads on the loç _{Cross-structure, irzcreasirrg'ship} structure and weight, but can also induce iibratiorrs and accelerations

resulting in serious

structural fatigue problems. ecent experience suggests that methods developed for monohzzll slamrdrtg are inadequate for SWATH ships.

ThIs should come

as no surprise since the noriohull method was developed to

pre-dict sLams on the ship's bottom: whereas, SWATH Impacts take _{pace on the upper}

structure connecírig the two struts. _{Obviously, new tools are needed by the}

designer to establish conditions under which wave Impacts occur, and to

An iiportant indicator of a sMp's susceptibility to slamniing is the noether of occurreeces of slzimdrig per unit tíie. This not only provides a relative

measure of

the merit of different STJAIH designs fron the perspective of slamning, but also helps to identify those factors influencing the ship's

sla=ing characteristics.

The

occurrence of a cross-structure slam Is dependent

upon at least three conditions. They are:

I. Entry of the cross structure Into the water (An obvious requirament.)

An entry velocity exceeding some threshold velocity.

A si'fl. angle between the cross-structure and free surface at point of impact.

As can be seen frein the above criteria, the relative motion between the

peint of impact arid

the

free surface is an important parameter in determining

the occurrence of a cr055-structure slaza. if it is assumed that the relative motion and the angle between the cross-structure arid free surface at the point

o f i mp.ac t a re s ta ti oria ry Cuass Ian processes, t hem the number of si aus per unit

of time can be determined from the computed statistical properties of these variables. The above assumption is reasonable in the context of linear ship

motion theory even thuh in reality the impacts produce sharp nonlinear peaks

in the acceleration. These impacts occur in a short interval of time arid the effect upon the ship displacement and velocity are minimal.

Since it is reasonable to expect that a more precise determination of the relative motion would provide a more accurate estimate of

the

occurrence of slxmirrg, a procedure vas developed for including

the

ship-generated waves amid che diffracted waves

fri the

coorpitatíons of the relatíve motions. Routine com-putatiorts only consider the incident wave as part of the free surface and

neglect ship-generated waves and corresponding diffracted waves. Computations were also made to determine the effect of assuming that impacts occur for a limited range of angles between the deck and the free surface at the point of

impact. This also has not been considered la routine computations. BACKGROUND

A method for estimating the expected number of slams per second of a

mono-hull or conventional ships bottom was developed by !ick', based upon the

rela-tive bow motion and angle between wave and keel at the contact point. lt was

:»

assuzited that the slaai occurred when the relative velocity between the bou ai

the sea surface exceeded a critical aniomt at the tizne of contact, the bou ce

out of the water previous to contact, and the angle between the wave and keel at

se chosen contact point vas small.

In addition, the relative motion and angle

between keel and wave were assed to be stationary randora Cuassiart processes.

arrived at the se formulation, excluding the effects of liimiting the

angle between wave and keel, by assuming a

ore restricting narrow band process,

arid was able to obtain other important statistical properties of the slaanirtg

phenomnezion.

_{Ochi was also able to derive empirically a threshold velocity of 12}

f tfsec (3.7 nt/sec) for a 520 foot (158 meters) Mariner Class ship f

ro

nodel

experinments in irregular waves.

Ochi's data are show in Figure 1.

In a

sub-sequent paper, Ochi3 proposed that the threshold velocity of 12 ft/sec (3.7 nt/sec)

found for the Mariner Class be Froude scaled for ships of different lengths.

This is the general practice currently in use for computing the

expected nuzitber

of occurrences of nonohull bott

sllng.

Application of the above criteria to the botton of the deck structure

con-necting the twin hulls of a SAIH ship has not been verified and

_{a procedure for}

scaling the thresbold velocity fron the botton slannning of

a Mariner Class hull

to the cross-structure of a SWAIH ship is not readily apparent.

Figure 2 sbous

a plot of the cross-structure slazmaing pressure variation with respect to the

relative velocity obtained fron model experiaents with a 1/32 scale

SWATH T-A()S

in waves.

_{These data show sic occurring at relative velocities}

_{as low as 1/2}

ft/sec (.15 nt/sec) at a ship speed of 3 knots in head

waves and as low as 2

ft/sec (.60 nt/sec) at a ship speed of 8 knots.

The threshold velocity obtained

by Fraude scaling the value for the Mariner Class (based

_{upon length) is larger}

than the velocities at which slants were recorded for the T-ACOS tirodel.

Another

paradox is that the expected number of occurrences of slazrmdng conputed frani the

relative notions arid the observed threshold velocities do

_{riot coincide with the}

actual tneasured values.

_{It is guite evident that the conventional formulation}

used in estimating the expected number of occurrences for nonohulls

_{is not}

directly applicable to the SWATH cross-structure problem, and the influence of

other parameters such as the angle between the structure and free surface

needs

...

M,

T

_:'

r

PROBABiLiTY OF SLAMMING ThCLITDINC A1'iGLE OF IMPACT

Chuang4 developed a relationship between pressure and velocity for enti-mating maximum slamming loads ori high speed craft which is given by

Max pj 12V

where k is an arbitrary constant

P is the mass density of the fluid and

IT is the velocity normal to the wave surface

The impact pressure p1 is that part due to the velocity component of the craft normal to the wave surface. The total impact pressure includes a contribution

due

to the forward velocity of the craft. _{Of particular interest is the fact} that the constant k is actually a function of the impact angle. i.e., the angle between the structure and the free surface at the point of contact. Figure 3 presents the relationship between the constant k and the impact angle as established by Chuang. _{Chuang5 has applied this method to the cross-structure} slamiririg of a catamaran with good results. In this case, the slamming pressure due to the horizontal velocity component of the ship could be neglected without serious error because of the relatively loz speed of the ship. Since the

cross-structure of the catamaran is very nearly the s as that of a SWATB ship, it can be reasonably assumed that the expected number of occurrences of slamming per unit tine is also dependent upon the angle of impact.

As previously indicated, Tick developed an analytical expression for

cori-puting the expected number of slams per unit of time of a ship's bottom based on the conditions that at the time of impact: the

relative

motion Zr(t) passes through the value -k; the relative velocity Zr(t) exceeds none threshold velo-city v>O; and

_FF.í<.

where is the difference between the angle of the keel and the slope of the wave at the bow. It is assumed that the relative motion Zr(t) and the angle (t) are stationary Guasian processes.

The development is

a generalization of the method of RiCeb and establishes the probability of an level crossing in the inter-val between times t, and t + dt, under the specified conditions. In the following notation, x1, x2, X3

correspond to the relative motion, relative velocity, and relative angle, i.e., w1 _Zr(t) x2 Z.(t),

r

«t).

For dt

sufficiently sa1l x1(t) can be

considered as a straight line

in the

interval and

x1(t) _{- x} (t3) +

2

(t3)(t - t0)

Then, if p(x1,x2,x3) is the

joint

probability of the randoc variables, the

pro-bability of a k-crossing with the required properties in the interval dt is

given by

=

f(dt)

dt

₃

f(x, 12, 13)

"

3-x2dt

and since the irttegrand is continuous this reduces to

2

f(dt)

-dfdx3Jdx2x2

f(10,

X,

13)

for Gaussian variables

f(x1, 12, 13) 1

exp(WQ)

(21»312 Dh12

where

Q

:cTijXizj

is the

elenent of matrix inverse to he matrix (ajj) of the covariances and D is the Determinant of the matrix

Integrating the above equation for the case

o2 =

over the interval from O to T gives the expected tunber of slams in the interval.

Dividing by T gives

the expected number o slams per unit time

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,,r

plb aM yai

tiot are

li&-r by

fa the forvard ape4 of the VATØ ahí p and

fa the encounter frequency.

The au-rge voce-nt fai,

_,

fa aaat

to be zero.

The

1itfôri

of 4j (J -

1, 2, 3, 4, asid 7) baa been preaented by

(n,)

(y,z;n,)di j'-2, 3, 4, 7

()

vhere

fa the source strength dfctrfbuted ou the

ATh ship's contour and

fa

t-1nefona1 creen function due to a unit source on the contour

The Creen

f n tión, C, Is giren by 1ehatzaen and Lai

'-

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- 1og(y+1z-iw1)

+ 2

-ihere b',z

la the point ihere the potential f. sought and (n, ) is a source

point at the SWATh .hlp' a contour.. The source strength,

fa deteriifned by

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- -iri, for

J

2, 3, 4

(10)

,

for

_J

1* the

ip&nent of the unit nrma1 'ector directed into the fluid.

11

The absolute i't-ttical

otioii of the

IAT1 ship f. cputed by

-

4- y4 -

(11)

The relaeie bo

zotion (ZM) is sfly the difference between

atfons (5) and (11)

-

+ y

x)

-In the nuerfcal côntatf on of 4uation (12),

the aplitude of

uotfon, ci,

is cçsxted with SWATh ôtfii progra

and the velocity poential,

f.

c-puted with WtTRO2 (ship iiotion prograzn) at

_{a gí'ien point (y,z)}

located ouC;fde

the wetted bOdy.

Ciparison with Ezperfnents

A copaxf son of the coaputed

relative øotfons with ezperfenta1 results for

three lc

f oms ori T-AiS SWATH i. presented fri FIgure. 3 through 1(.

The

location. corrspond to the forward, aidshfp and after portion of the

cross-structure along the center line of the ship. Figure 3 shi's the coputatfous

of relative

tbon in head waves at a shf p spee4 of 3 biots with only the

md-dent vve representing the free surface and again with all three

coiiiponents:

incident wave, ship generated wave, arid diffracted wave Included. These results

shw chat the coputatforns vi th Just incident waves are very close to the

experi-ental results and paradoxically, including the ship-generated

wave and

diffracted valles, fri general, degrades the correlation between the co«iputed and

eperírental result..

The zaae trend is evident at a heading angle of 135 degre.e

and a speed of 3 knots as shown fn

Figure

9..

In beari vives at a speed of 3

knots, Figure 1C, the incident wave vas the doninant free surface factor and the

ship-generated waves arid diffracted wave corponerits had ari insignificant effect

upon the cop'uted results.

It is interesting to note that Lee, et al.,7 in their re exteisive studies

of the relative notion conputations of co1iulls

concluded

that there is no

conclusive evidence

that the inclusion of diffracted and notion-generated waves, as conprited by strip theory, inproves the results. Apparently, the s

conclu-sin can be

nade in

regard to SWATH-type ships.

StAZT AXD COCLUSIOS

An rigatfn vas nade to mnprove nethods for estinating the expected

number of occurrences of SWATH ship cross-structure slannløg per unit tine. Slamming of che cross structure Is an inportaut consideration in the assessnent of the operability of SWATH ships in waves and the utther of occurrences per unir

cine provides a

quantitative ncasure of the slamnlng characteristics of

SWATH ships. A nethod fr conpucíng the mber of occurrences of slans for

nönohulls fron relative ship notion has been in use by the ship designer for nany years. lt is essentially a level-crossing problen based upon the

asstp-tin char a sian occurs when the

keel

at the point of inpact, enters the water

with a velocIty greater than sone uniting threshold velocity. The threshold

vlocíty has been deter'nined fron nudel experfnents for a Mariner Class ship in waves. This threshold velocity is Fronda scaled for ships of different lengths.

The same approach is dfrcctly applicable to the cross-structure slaing of

SWATH ships. -er, it is not readily apparent how to scale a thri,shold velo-'ity value for keel slaing of a Mariner Class ship to the cross-structure

slanrIng of a SWATH.

In addition, it is believed that other paraneters such as the inpact angle nay also gci7ern the occurrence of a slan on the SWATH cross structure.

Calcula-tion of the expected number of slans for a SWATH T-ACÛS fron the relative noCalcula-tion using the lowest Imnpact velocity recorded, as the threshold velocity resulted in a value niich higher than the observed value. It vas concluded that other

para-neters,

such as the uniting impact angle, contributed to thís discrepancy.

Calculations

were nade showing the

extent to which the uniting fnpact angle

reduces the estinated nunber of occurrences of slams, but there was insufficient experine-r-mcal data to definitively define a uniting impact angle and, therefore, the results of this aspect are inconclusive.

13

Since the relative notion

is

an important parameter

for estinatf

ng the

f

re-quency of slaing, an exa1nation was iade to deter1ne the effects of

including ship-generated waves and diffracted waves

in the

cputation of the

relative notion. Strip theory vas used to coute

the

ship-generated wave arid

diffracted wave which vas added to the incident wave in describing the free sur-face away

fron

the hull at the point

of furpact of

the cross structure.

Copu-tatfons nornally include only the Incident wave.

The results of this frwestfgatforz showed tb-at there vas no Itnproveiment in the

co-nputed

relative notion (when con-pared to expermnental results) by the

Inclusion of the sb-f p-generated wave arid diffracted wave In the free surface

elevation.

In the cases exaialried, the ase of the incident wave alone gave

better results or the sane as that obtained by Including the ship-generated wave arid diffracted wave.

In sary, che following conclusions can be nade based o-ri these studies: The frequency of slaiiimlrtg of SWATH cross-structure is not only dependent upon the relative notion, relatIve velocIty, ar4 a threshold velocity as fri the case of uionohull keel slaimning, but is also dependent

upon so-ne uniting Iriipact angle.

There is insufficient experinental data to establish lfiting Inpact

angles for SWATH ships or other paraneters influencIng the frequency of

slnnIng.

The inclusioni of ship-generated wave and diffracted wave in the con-putation of relative notion, which is needed for the estination of the

frequency of slarinIng, do-es riot in-prove the correlation with nadel

experinent results.

lt can be

concluded

fro-n the above that additional nadel experinents are required on a SWATH ship to obtain data to specifically

address

the problen of estinatinig the expected nunb-er of inpacts per unít of tine by obtaining, in addi-tion to the custonary neasure-nents, other inportarit nesureitents, such as che

angle of liipaot.

Tick. Leo J..

Certain Probabilities Associated with Bow Su1rgence

and SMp Slazning in Irregular Seas,

Journal of Ship Research, Vol 2, No 1, 1958.

Ochi, M. K..

Predíction of Occurrence and Severity' of Ship Slaing at

Sea,'

Fifth Symposiuzx on Naval Hydrodynaiics, Office of Naval Research, ACR112,

1964.

0cM.. M. K. and Motter, L. E.,

Predictfon of S1Ing Characteristics

and Hull Responses for Ship Design,

Trans. S!AME, vol 1, 1974.

Chuang, Sheng-Lun,

Slamnfng Test of Three-Dimensional Models in Calm

Water and taves,

SRDC Report 4095, Septeiber 1973.

Chuang, Sheng-Lun, et al.,

Experiental Investigation of Cataaran

Cross-Structure Slazxmdng,

NSRDC Report 4653, Septeeber 1975.

Rice, S. O.,

Mathematical Analysis of Rando %oise,

Bell Sys. Tech.

Journal, vol 24 arid 25, 1944 - 1945.

Lee, Choung M.. et al,

Prediction of Relative Motion of Ships in

Vaves,

Fourteenth Syinposimi on Naval Hydrodynamics, Office of Naval Research,

National Acadetny Press, 1983.

Frank, W..

Oscil1ation of Cylinder in or briow the Free Surface of

Deep Fluid,

David Taylor Model Basin Report 2375, October 1967.

Uehausen, J.V., and Laitone, E. V.,

Surface Waves,

Handbook der

Physik, vol 9, pp 446 - 778, 1960.

15

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Figure 1 - Mariner Class Slai

_{Data (Ochi2)}

200 100 80 60 40 20

io

-40.0 20.0

z 8.0

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LATIVE VELOCITY IN }Tïi/SECCD

t e

i

t I I J

0.12 0.2 0.4 0.6 0.8 1.0 2.0 3.0 RELAiIVE VELOCITf IN NETtRS/SECOND

FI gure 2 - Cross-Structure Slaia Data for T-AßOS SWATH

17 SHIP

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Figure 5 - Relative Motiork Response Operator for TAGOS at 3 Knots

in Head Waves

20

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