Analysis of slamming phenomena on a model of a cargo ship in irregular waves

Technische

Dell

Ref.: SI 111/29

ANALYSIS

OF

SLAMIING

PHENOMENA

ON

A MODEL

OF A

CARGO SHIP

IN

IRREGULAR

WAVES

by

PERDINANDE

Laboratory of Naval Architecture

University of Ghent, Belgium

-

196,8

ANALYSIS OF SLAMMING PHENOMENA

ON A MODEL OF A CARGO SHIP

IN IRREGULAR WAVES

by

V. FERDINANDE

Laboratory of Naval Architecture, Univeraity of Ghent, De1giu.

Auguet 1968

Ceberena

21,rue des Draplers

Ref.

S 111/29

SHIP IN IRREGULAR WAVES

by V. Ferdinand.

Summary

Model tests in irregular head waYe were analyzed with special mpha.i.s on the slamming phenomena, endured by a

cargo ship in the ballast condition. The condition.

leading to slamming, i.e. the magnitude of vertical relative

motion. and relative velocities a.t the forward perpendicular,

weT. evaluated. _{By means of thCoretiCal computations of}

pitching and heaving motions and their phase angles in

regu-lar waves, relativ, motions &fld relative velocities _{could be}

evaluated at any station aft of the forward perpendicular..

A distribution of the larnming deCeleration., leasured at a

certain station near the bow, allowed _{one to determine the}

so-called tmthreshold velocityu, figuring in the _well-known

probability formula, which lets one predict the occurrence of

slamming. These throshold velocities _{are calculated for}

diff.rent stations along the ship length and for different

level, of slamming severity. _{It Li noticed that these}

threshold velocitie. an dep.ndent on ship speed. The

meaning of the cOnception Nthresbold velocitr is discussed. In order to predict slammmg, as recognised on a ship in bad leather, au indication concerning accepted slamming sevinittee

is given. _{Estimating the number f slams in a hnndr.d motiam}

Oscillation, which ii likely to be accepted by a ship's

master, the attainable speed in the here investigated sea

ntroducti.n

Ship. actions in regular waves and their phase angles with respect

to the wave can be calculated or derived from model tests, Tb. knowledge of pitch, heave and their phase angles lets one calculate the vertical motion, velocity, acceleration and the relative vertical action and velocity with respect to the wave surface at any point along the chip

length. Uencs, by using the superposition principle, the variance of

these can be determined for the ship sailing in any irregular sea the

energy spectrum of which ii known. In particular, the knowledge of the

variance of the relative vertical motion and velocity at a station near the bow is of interest for the prediction of the occurrence of slamming (and of the shipping of green water aa well), following the method of

Dr. M.K. Ochi, (ref. 1). This prediction is of great importance because

it makes the judgement of the ceakeeping abilities of new, fast ship types possible at the early stages of design

The result of this probability calculation depends on the variances

of relative flotion and velocity, on the draft at the bow, and also on

th. value of a factor, called 'threshold velocity', i.e. a relative

vertical velocity betweeD the wave surface and the bow of th. ship. There may arise some doubt while adopting a certain value of the

'thres-hold velocity'. The choice of a 'threshold velocity' for a particular

ship, based on values which were used by some authors for other ships, might be hasardous at this stage of invistigation. More information

about 'ttreshold velocities' for different kind .f ships ought to be gathered in order to provide a general and adequate guidanc. to ship

designers. It is the purpose of this.report to add a contribution to

the bulk of information which is published already by other anthors. On the other hand, it is not sufficient to predict the physical appearanc, of a' slam, i.e. the sudden building up of. bydronamic

suree ofl the shp'o bottom at the inatant of contact with the

wave surface. A captain does not respond to slams under a certain

level of seyerity. Hi response of course is quitO individual.

How-ever, the decigner ought to have an idea of this level of severity and of the number of such slams in a hundred low cycle oscillatiOns a

captain is ready to accept before reducing the ship speed.

Model tests in irre&ular EeaO

Model tests in irregular head waves wOr.e carried out at the

Davidson Laboratory (StevenS institute of Technology) on a 1/96 scale

model of a cargo vessel of the type "LU-ship" ("Compagnie Maritime

Beige"). _{These Eodel tests are described in ref. 2 and 3. Some main}

The ship has a raked stemand a cut-away forefoot. _Additional

iflfor-mation and linea are given in ref. 4 and 5. _{As thO area of the flat}

forebottom certainly is an important factor which ought to be

consi-dered in thi study of slamming, the breadth of the flat bottom versus ship length is given in fig. 1. _{Among the model teSts, only these in}

the ballast condition are analyzed here, _{because slamming did}

_not

occur in the loaded condition. _{Tb. test conditions (full-scale}

dimen-sions) were :

characteistics Of the ship are

Length between perpendiculars, L, metres 136.00

Breadth moulded, metres _18.70

Depth to maindeck, metres _9o45

draft at the forward perpendicular, metros 4.25

draft at the after perpendicular, metres 6.10

freeboard at stem, metres

1475

diaplacement, metric tone b950

block coefficient 0.66

longitudinal radius of gyration, metres 33.50

longitudinal moment of inertia, ton m. sec2 1 025 000

longitudinal position of the centre of

gra-The energy spectrum of the irregular waves was calculated by

means of the Tukey autocorrelation method, The average apparent period was calculated according to the formula

z.

1>t

-where is the energy density at a certain circular frequency

The wave spectrum is shown in fig. 2. The area under the spectrum

curve is twice the variance of the wave record.

The main characteristics of t irregular wave pattern (full-scale

dimencione) were

4

This irregular wave pattern corresponde to a fully-developed Beaufort 7 sea state according to Neumann and to a Bsaufort 7 - 8 sea state

E twice the variance of the wave record, sq. mótres 5.75

Significant wave height, metres _6.8

- average appareut period, seC. 8.7

vity CO., aft of the midship section, metFos .1.80

still water natural pitching period.,, sec 6.19

accordiug to the Pierion - MOakOwitz curve.

The model was run at different apeed, corresponding to

ship

speeds of U.21, 8.53, 10.82, 13.10 and 15.43 knots.

Pitching and heaving motions, bow.acceleration, relative motion

a.t the bOW and wave height were recorded simultaneously.

The accelerometer was placed at 0.169 L aft of the forward

per-pendicular FP. The TMrelative motion wirOw, a resistance wire gauge,

was fixed 1/2 in. (1.2 m. full scale) forward of the stem and parallel

to the stem rake. A wave wire gauge was installed on the carriage

5 ft. (146 m full-scale) forward of the C.U. of the model.

Analysis of the model tests

It is assumed that the random variables pitch angle, heave,

ver-tical velocity, verver-tical acceleration, relative verver-tical motion and relative vertical velocity at any point along the ship length have a

normal distribution with zero mean and that their spectra have a narrow

frequency band. _{Hence, the Ndouble amplitUdes" of the variables, i.e.}

the. readings from a maximum (peak) to a minimum (trough) of the

oscil-lations with a zero crossing have a Rayleigh distribution. As the

recording of these variables was kept up during 120 tO 150 oscillations

for each ship speed, a derivation of the variance from the

Loflguet-Higgins formula

AvgX ₌

_l.77Jx

₍₁₎

seemS to be fairly accurate from a statistical point of view. In this

formula i.e

AvgX _{the aVerage of the double amplitudes of theviab1e} I,

If the variable X is the derivative of the variable X , the xpect.d circular frequency Li given by

(a)

E is derived from the records of vertical acceleration at

aACC

station ACC (0.169 L aft of the FP) by means of formula (1). The

zero-line of these records is exactly known. The area which is bounded by the zero line and a recorded half oscillation under (above) this

zero-line, is, after scaling, the range between a maxirnum (minimum) and

a minimum (maximum) of the oscillation of the corresponding vertical

velocity at station ACC. (July the ranges with a zero crossing ought to

be retained. The existence of a zero crossing is easily verified by

an investigation of the simultaneous pitch and heave records.

_EACC

for the vertical velocity at station ACC is derived from the average

of these ranges by means of formula (1). This average, provided a

proper scaling, is obtained in practice by déviding the sum of all ordi-nates at equidistant time intervals of the acceleration record under

(above) the zero-line by the actual number of oscillations. The

expec-ted circular frequency for vertical acceleration, velocity and motion at any point along the ship length is then

L.)

according to formula (2). The.accuracy of this procedure and the

assumption of a narrow-frequency band spectrum can, be verified by a

comparison with the average period T - of the acceleration record.

The difference 'between the calculated _{expeotedu circular frequency} (

and .) appears to be not higher than 1 per cent. _ESACC for the

ver-tical motion at station ACC can be given by

EACC.

.ACC

On. can express, by approximation, this vertical motion at itation ACC

in the form of a sum of the pitch motiou at ACC (pitch angle in

radIans z distance from c.G.) and a fraction of the heave motion

V 'sACC -

'/ACC +

(3)

wheil - , if 1 is the distance from C.G. to the

con-sidered station.-

fi

and are derivàd from the pitóh angle and the heave records respectively by means Of formula (i) m is derived

immediately from (a). This approximation is equivalent to assuming the

vertical motiOn at the station ACC being related to the rotation around a virtual "centre of rotation", at a distance 1' aft of the FP. 1'

can be derived from v/'TCC and

/c.

lience, the vertical motion and

velocity at the FP can be evaluated too, by means of the following

expressions

= + m or - 1' and.\/Fp

No significant difference between the average periods of motions and.

relative flotion at the FP can be stated--on the records. Hence,

ver-tical motiOn and ita derivatives, an4 relative verver-tical motion and its derivatives have about the same "expected" circular fiequeflcy W0

for the relative vertical motion at the FP can be derived from

srPP =

the records by formula (i), and for the relative velocity at the FP

2.

En

vrFP ° _arFP

Table I gives the values of the differint E, ci, , m and 1' for 4

dif-ferent ship speeds.

Slamming was mild and not frequent at thi speed of 6.21 knots,

and therefore. this case is not considered in the further anal.yiis.

In order to evaluate the variation of E and E along the yr

in,. regular wav and their phase angles with respect to the waves was carrild out for the ship speOds 10.82 knots and 13.10 knots

as well, The results, which will, be used in the discussion on the

"threshold velOcities', are given further down.

The theoretical prediction of the occurrence of s1ammin

According to Ochi (ref. 1), the probability, of slamming is given

2. 2.

E

Prob ,[ s1am

L

,

' (4)

by

where draft at a given station near the bow,

vr the "threshold velocity" at this station.

E , E twice thevariance of the relative motion and

or yr

the relative velocity at this station respectively.

In order to be in accordance with the symbols airOady used in the present wOrk, the symbols in (4) are somewhat differeát from those

used 'by Ochi.

Forula (4) ii valid if "the relative motion is cosjderad as

a random variable having a narrowband normal djstrjbutjon with zero

mean", (ref. 1). _{In this caie, the probability density function of}

the relative motioO is a Rayleigh distribution.

2.

4

_.1-'c

(5)

kiere, ir is the value of peak (trough) Namplitudes. Fig. 3 shows

these probability density functions for E5 30, 40, 50 afld 60

The area under each curv, is unity and the probability of exceeding a givin peak value of the relative motion sr0 _{is the area under the}

curve on Ui. rightbaud side of.. sr0,i.e. the integralof the

bility function from or0 to infinity

2. Prob [ or or.]

The assumption of a normal distribution with zero mean implies also

that

E

yr 0 _or

(7)

Formula (4) is based on the following theoretical conception : a slam

will occur if the relative motion exceeds the draft, and moreover if in thi8 case the relative entrance velocity at the instant of contact of the forebottom with the wave surface exceeds the "threshold velocityu

vr

Because of relation (7), (4) can be written in the form

HL

P

- £

where

Slamming will occur if the relative motion exceeds a certain value

U', (fig.

3).

Formula (4) can also be written in the form

,2.

P [slam

- -Q

E

where

Slamming will occur if the relative velocity exceeds _{a certain value vr'.}

These last two theoretical conditions and the condition at the

basis of formula (4) are identical.

The above mentioned assumption of a narrow frequency band implies

that a relativ, motion oscillation has a(nearly)sinusoidal form. The

amplitud, has to exceed H and has to reach at least U' (sr ii'

iut)

in order to obtain a relative velocity _{vr (. if't0 coo tt) at the}

not strictly the meaning of a minimum Value of relative ri-entry

velo-city inducing slamming.

r 10

moment the relativo otion becomes II _(.. II' sin (J0t) again. Frosi

the two last m.ntion.d relations one derive. immediately

LI, which is relation (9).

Formulae (4), (8) end (lu) are related to one, properly chosen station along the ship lemgth.

The magnitu4e of the hydrodynamic impact force, and thus the hydrodynamic pressure at this station determines the severity of

slam-ming. This pressure is supposed to be proportional to the square of

the relative re-entry velocity. To any degree of slamming severity

corresponds a certain 'threshold velocity' and hence, formula (4) can

be used to predi.c.t the probability of slamming above a given degree

of eeverity, if'the proper corresponding threshold velocity is known. In the same way as indicated above, formula (4) can be transformed into an exprOssion containing only relative motion, and where IP. is

the prerequisite amplitude of relative motion for a given degree of

slamming severity.

cA7c4t

It must be noted that H' is related to a givenH (while _vr

18 not), an4 such transformation will only have sense and be useful for the evaluation of the threshold velocities by the statistical

analysis of. the model teete.

The word 'theoretical' in the former lines was used deliberately, becaus, the further statistical analysii indicates that, in th. case of this Lu-ship, the 'threshold velocity', as used in formula (4), ha.

The distribution of the measured slaamin& deceleration.

The e.verity of slamming on a given ship i. proportional to the magnitud. of th. deceleration peak, which appears on the bOw aàcele-ration record at the istaut of an impact. This magnitude is measured from the peak point to the drawn continuous trace of the low cycle os-cillation at the same point of time.

Fig. 4 shows for the 4 ship speeds the histograms of the slam decelerations at station ACC, in per cent of the number of relative,

motion oscillations. The percentage of oscillation8 without Slam

deceleration is jndjãated also. There is some doubt of this latter percentage because the interpretation of a very small deflection in

the acceleration record can be difficult. The number ofdeceleratione

under 0,lg, for instanc., seems to be a more reliable figure.

TherS-fore a. cumulative distribution of the slam decelerations for each ship

speed is calculated. _{Each plot on fig. 5 indicates the percentage of}

rlativo motion oscillations with a deceleration peak (including _{0 g)}

under the correeponding deceleration level. The curves through the pies represent an estimate of the cumulative distribution of Slam decelerations for each ship speed. _{From fig. 5, one derives the} fre-quency of occurrence of slamming with decelerations above a certain

level. These numbers per 100 _{relative motion oscillations are given}

in Table II.

The determination of the 'threshold velocityN

The relative lotion dOUble amplitudes have a Rayleigh

distribu-tion. _{Mince, the probability of exceeding a given (positive) value of}

'aaplitud. at the VP, is given by formula (8). _{The values for}

/

ii

tb. prObability .f occurrence of a relative notion aplitUde at the

PP ezc..ding a value indicated on the abscissa. Tb. probability of

slamming can. be expreseed by formula (a) where If' ha. the

YllUir

given by (9). Considering a certain lev.j of severity, a proper value

of _vr baa to be intrOduced.

The frequency of occurrence of alamming for several degrees of severity, as counted on the records and derived from fig. 5 (Table II) can be regarded as the probability of such slamming. The plotting of these values of Table II on the probability curves for the relative

motions at the FP in fig.. 6 lets one read off the values of 11' for

each ship speed, corresponding to the different levels of slamming

severity. These values of Ii' are given in Table. III.

2. 2. 2 Henci, as E E. E ar sr yr or, according t (9), v - (u.' 112) (9')

the corresponding values of _vr* can be computed. Fig. 7 shows the

values of vr for the different ship speeds and for the different

levels of slamming severity. The linearity is noticed, except at

the highest levels of slamming severity. _{This seems to confirm that,}

up to a certain degree, the severity of slamming is proportional to the square of the relative reentry velocity. _{The plotting in fig. 7} rev.ala also that the "threshold velocities" are dependent _{on ship spó.d.}

Finally, fig. 8 shows the "threshold velocities" at the forward perpendicular versus ship speed for each level of slamming severity.

Curves were drawn through the plots at the speeds of the investigated

runs. The lower the ship speed, the higher the "threshold velocity"

(for th. same s.v.rity of slamming) seemi to be. Hence, th, probability

de-creasing ship spied than the deorsase of relative motion. and

veloci-ties (E and E ) alone would let foresee.

sr yr

There seeps to exist a minimum thr.shold veloCity however at a.

speed betwein 13.1 and 15.4 knot..

The thre8hold velocities in fig. 8 refer to the station at the

F?. The variance of the relative motions and velocities varies along

the ship length. At another station, aft of the PP , other values of the Nthreshold velocities" will be found. The wthreshold velocityu

yr

L at station o.x L aft of the FP is related to vr+ (at

the FP)by

2 2 ii2 ₂

+ o.,x L + vr

E

E..

E E

ST vi

sroxL

vroxL

since the probability formUla (4) ought to give the same result.

o.x L' vr L and L are twjce the variance of the

relative motions, of the relative velocitieB, and the draft at itation

o.x. L respectively, it i.e evident that also at station o.x L

2

L) ..E

vro.xL

°

sroxL

Hence, the following relation 1. valid

E E

2 ar o..z L 2 2 / 2 si o,,x L

-yr

vi

- L

H. _{- )} (12)

*o.x L 1*-

0.1

L- E.

sr sr

at a certain station o.x L aft of F? ought to be

calóu-sro..zL

lated. For a given sea spectrum, this can be done using the

superposi-tion principle, if the Chip mosuperposi-tions in regular wavea are known..

The computation of pitch and heave motion in regular waves with a given wave length

A

and an amplitude of 1 .etre for instance give.

14

The phass angles refer toa position, of the nodal point of the rising wave surface at the midship section, a the time t u o At this

in-stant, one can'writ. in complex form

O-a+ib

and

ic+id

The wavesurface elevation at the FP is - coB

1800

where - - 90° +

If the location along the ship length is indicated by the distance from

the midship section x'L(- 0.3 L - o.x L), then thó 'phase angle of the

surface elevation at a given station is

3600 - 90° + '

and the surface elevation in complex form is

-

, 3u0° . , 360°

= cos (-90° + x ) + i sin (-o° + x )

-

-

, 3609 . 360°

or 'y= sin x - 1 COB K

The vertical motion at station o.x L is

[(x' L + e) a + c,+ \(x' L + e)

b+

d i

where e is the distance from the CG to the midship s.ction (positive

i.f CU lies aft), and the relative vertical motion is

r , , 360°

ar

_L(1

L + a) a + c - sin x

360° ₁

L + e) b + d + cos x (13)

Iormua (13) leti one calculate the response amplitud, operators corresponding to different value, of the wave length _{A ,} and _E5

o x L

is derived from the ar.aunder the curve of the relative motion spectrum

at the conaider.d station.

The pitching and hsaving motions of the mLu-ship in regular

head waves, and their phase angles with respect to the wave were

wer. thos. already mentioned in th. present work. The method of

computation was based on the atp theory of KorvinKroUkovsky, (ref. o). The integration along the ship length was carried out by Simpson's

rule for. 10 equidistant stations plus an intermediate station fore

and aft. The damping force per unit vertical velocity on the ahip

sections was calculated using the Avalues as given by (irim in ref 7.

The damping force of the wave on the ship was not neglectable and was

incorporated in the computations.

The results of the calculations are given in fig. 9, fig. 10 and fi. 11; Negative phase angles are lags. Phase angle for

pitching means that the pitch angle is maximum WUpN at the moment the

nodal point of the ri8ing wave surface is at the midship section. Phase

angle

90°

means that the heave of the ship is maximum up" at

the moment the wave crest is at midsection.

Available results of model tests in regular waves of a length equal to the ship length (ref. 2) are plotted in fig. 9 and 1O for

COmparison.

-The knowledge of this theOretical values of pitching and heaving motion, expre8sed in complex form, allows one to calculate the response

amplitude operators for the relative vertiCal motions at different

stations along the ship length. _{The results are given in fig. 12 and}

fig. 1i -foi the two considered ship speeds. _{After transformatIon of}

the Wave spectrum at a fixedpoint (fig. 1) into the encounteredwave spectra correiponding to the considered ship speeds, the response

spectra Of the vertical relative motion at the FP and at the stations

0.1 L, 0.2 L and O.3 L aft of the F? were determined in the usual

way. The area F _{under the spectrum curves were evaluated. The so}

derived values Of

fE

at the PP are 14 and 17 per cent lower

on th. records for the speeds 10.82 knots and 13.10 knots

respecti-vely (see Evalues in Table I). As the differences for pitching and

heaving motions, found in a similar way, are much smaller, one can presuàe that the theoretical calculation of phaseanglee might be less

satisfactory. Nevertheless, it is accepted here that the reduction of

E along the ship length with respect to _Esr at the FP, thus the

ratio in formula (12), according to these theoritical

Sr

computations, is realistic. i'ig. 14 shows the values of

i.e. the reduction of vertical relative motions along the ship length in the here cOnsidered irregular wave pattern. It is seen that there is only a small difference between the two curves related to the

dif-read off a value yr

ar o.x ferent ship speeds. The curve of the ratio

E

slamdeceleration curve u > g" in fig. 15.

.Jnvestgsti.n

of

recorded slams

The determination of "threshold velocities" at a given station

16

in fig. 14

is derived from the average and is aoeurned to be valid for the range

of all here considered ship ápeeds.

Finally, the "threshold velocities" at any station can be evaluated. Fig. 15 shows curves of "threshold velocities" at station 0.1 L and

0.2 L aft of the F1' versus ship speed, for different levels of

slam-ming severity, as derived from fig 8 by means of formula (ii).

According to Uchi (ref. 1), the 'threshold velocity" for a 520ft. ship ("Mariner") is found to be 12 ft/sec at station 0,l L aft of the

ji'l' _{The ship was tested at a speed of 10 knots. If the Froude. scaling} law is applied, this corresponds to a threshold velocity of 3.4 rn/sec

at a speed of 9.25 kn0ts for the here investigated "Luship". One can

location of impact along the ship length.

may be of practical interest, because it allows one to use the proba-bility formula (4) to predict the occurrence of slamming at an arbitrary

degree of severity. Actually, the location Of the station along the

ship length where slamming happens is random. ilence, a "threshold

velocity", introduced in formula (4), at a certain fixed, chosen station is artificial to some extent, since it implies that slamming always

occurs at this locutiOn. The use of this artificial "threshold velocity"

at station PP (where no flat bottom exists) is as appropriate as the

use of the corresponding (lower) "threshold velocity" at station 0.1 L

for instance. However, this "threshold velocity" must not strictly be

regarded as the value Of the relative velocity at the instant of impact, under which only a slam Of a lower degree or no slam can, and beyond which a slam Of a higher degree has to occur. The relative velocity

at the FP, at the moment of slamming, was derived from the records of

relative motion for the 5 investigated ship speeds. The slam

decele-rations were plotted versus simultaneous relative velocity in fig. 16. The wide scatter of the plots is not only due to the dependence of the

relative velocity on ship speed. Fig. 17 and 18 show the same plotting

for V - l3.l and 15.4 knots respectively. The traced curve

repre-sents the formerly derived Nthreshuld velocities" at different levels of

slamming severity. it is seen in these two figures that the measured

relative entrance velocities, at the same level of deceleration, vary

within a wide rangeo _{Thia can be explained by the randomneSs of the}

In order to locate the impact', aD attempt wal _{ds to find the}

positions of the ship on the wave profile at the momsnt of slamming. An Ipproximete position was found assuming that the celerity of an

ir-z

r.gular wave is -j-- of the celerity of a trochoidal wave, corresponding

18

d.riv.d, and knowing the distanc. of the wave probe to the YP , this

latter could be located on the time scale of the wave record.

Simulta-neous heave and vertical motion at the bow was derived from the motion

records The knowledge of the forefoot emergence or immersion at the

same instant allowed one to verify or correct the position of the ship

on th. wave profile. _{It had to be assumed that this profile did not}

vary during the laps of time in which the wave travelled from the

probe to the bow.

Though the assumptions might gIve cause for some doubt, it was

possible to distinguish three circumstances of slamming

- a slam can occur after a partial emergence. of the keel, the forefoot

being immersed already. _{The final contact of the bottom with the}

wave trough surface can be located at an appreciable distance from

the forward perpendicular;

- for sOme slams the location of impact can poorly be defined, the forebottom diving into the water over a certain length, from any

station to the forefoot;

- a few slams, of a lower degree of severity, even occur at a moment

the whole

keel and

the forefoot seam

to

be immersed already.

The exietence of different circumstances in Which ships can slam

is not likely to simplify the problem of predicting the occurrence of

slamming. _{A more thorough study of different kinds of recorded slam}

phenomena could throw more light on this problem and is under way,

ilowever, the use of the concept "threshold velocity" migh.t remain

jus-tified and make in practice a reliable prediction of _{slamming possible.}

Acceptab'e levels of lammjn severity

A captain does not respond to slams and does not' give _{order to}

degre. of sev.rity. Previous experience on a similar, but somhat

longer ship (146 m), the m.v. "Jordaens" (ref. ), lots the

experimen-ters believe that a whipping str.ss in the gain-deck at the midship

of 0.6 kg/sq.mm corresponds to a slam which might be taken into

account by the ship's master.

There i. a certain relationship between whipping stress and slam

deceleration. Fig. 19 shows a plotting of deceleration magnitudes,

measured by meftns of an accelerometer near the FP , versus whipping

atresses, recorded simultaneously at the moment of slamming on the

m.v. "JordaenO" in a medium-loaded condition.. The "threshold whipping

stress" 0.6 kg/sq.mm on the m.v. "Jordaens" seemS to correspond to a

slam deceleration at the FP of 0.15 g. Considering the longitudi-.

flal moments of inertia

_of

the "Jordaens" and the here investigated.

"Lu-shIp" in their respective loading conditions, one may accept 0.25 g

as a good estimate of the deceleration at the FP , which might disturb

a captain on the "Lu-boat" in the ballast condition. Taking account

of the location of the virtual "Centre of rotation" (Table 1), this deceleration peak is reduced at station ACC to 0.18 g , which would

be the level of slamming severity among those represented in fig.. ₈

and fig. 15, beyond which the shock is considered by the _{captaifl as a}

"real" slam. Several

slams of such

severity however have

to occur

be-fore the captain decides to reduce the ahip speed, in order _tolower

the risk of damage. _{The readiness to accept a certain number of slams}

beyond this severity is quite individual of course, but 5 in a hundred

motion oscillations might be a good estimate of this number.. _{Using the}

values of the "threshold velocities"

vr0

1 L for station 0.1 L aft of FP at the severity level . 0.18 g in fig. 15, and the values E

and E of Tabl. I, but reduced to their actual values _{at 0.l'L by}

yr

20

formula (4), the zpectOd number of slams above the severity level

0.18 & in a hundred motiofl oscillations in function of th. ship ip..d

This expected nuib.r of slams is shown in fig, 20. Hence, one can cOn-dud, that, for slamming, the attainable speed of the "Lu-ship" in the ballast condition, sailing in a head sea with a significant wave height

of 6.8 metres and an average apparent'period of 8.7 seconds, is about

8.5 knots.

Conclusions

'1. The value of the relative vertical velocity at the bow at the

in-stant of a slam of a given severity is not fixed, but varies within

a wide range.

.... An investigation of several recorded slamming phenomena leads to a dis.tincion of different slamming circumOtancea. The location along

the ship length of the impact is' variable.

i. The Nthreshold velocity", as derived here from the statistical ana-lysia of records, i.e. that value, which has to be introduced into the probability formula, in order that it may give the observed

frequency of occurrence of slamming, is seen. to be not a strict

mi-nimum value of relative re-entry velocity, under which no slamming

of the considered degree of severity can occur, and has to be

re-garded hers as a rather artificial concept.

To some extent, the squared "threshold velocities" increase in a linear way with increasing slamming severity, related here to

dece-leration.

The "threshold velocity" is seen to be dependent on ship speed. The

probability of slamming at a given degree of severity decreases faster with decreasing ship speed than the decrease of relative

The, hers derived '"threshold velocity" at station 0.1 L aft of

the PP , at the severity level >0 g , corresponds fairly with the

threshold velocity as,given by Ochi for the Mariner-sh.ip

Accepting a five slams beyond a critical severity level

(correspon-ding to 0.6 kg/sq.m whipping stress) in a hun4red oscillations1

the attainable speed of this "Lu-ship in the ballast condition,

sailing, in. a Beaufort 7 - 8 head sea (according to Pierson -

Mosko-witz), seems to be 8.5 kflots.

Acknowl edgment

The model tests were carried out at the David8on Laboratory (Stevens Institute of Technology) under the guidance of Prof. E.V. LeWis, and'sponeored by the SOciety of Naval Architects and Marine

Engineers. The suggested critical aeverity of slamming is derived from the results of full-scale tests, carried out by theLaboratory of Naval Architecture, University of Gbent, under the directorship of Proc (ìe Aertssen and under the auspices of the Centre. Belge de

Mecherches Navalsa (CelisReNa). with the financial assistance of I.R.S.I.A.(Insti-tut pour la'Recherche S'cientifique dane 1' Induetrie et l'Agriculture).

9sf ersncss

Michel K. Ochi : "Prediction

of

occurr.nce.and severity of ship

8laming at

_see",

Fifth Symposium. on Naval Hydrodynamics, Offic.

of Naval R.iearch, U. S. A. and Skip aodelltank.n, Norvay, Berg.n,

September 1964.

V. Ferdinand. : 'Analysis of aodel tests on the LV. Lubuzba.hi in rou1ar and irregular waves", thesis, 1960, Stevén8 institute

of Technology..

V. Ferdinand. : "Model tests in regular and irregular

war.'

at the

i)avidson Laboratory, Stevens lnstitute of Technology,, U. S. A.",

appendix of ref. 4.

ti. Aertssen : "Service-performance and seakeeping trials on a.V.Lukuga", Transactions It. I. N. A., 1963.

(o)

i. Aertosen : "Sea trials on a 9,500 ton deadweight motor cargo liner", Transactions 1. N. Ai., 1955.

5. V. Korvin-Kroukovsky and Winnifred R. Jacobs : "Pitching and heaving aotions of

aahip

in rogularwavOs", Transactions

S. N. A. U. E. 1957.

(iris : "Die Schwinguugen von schwimmenden, zweidimensionalen

Korporn. Berechnung der hydrodynasiochen Krf to", Bericht

Nr. 1171, Deutsche Forschungsgemeinschaft, 1959.

(B) G. Aertsssn : "Servico-perforsance and seakeeping trials on

m.v. Jorda.ns", Transactions ft. I. N. A., 1966.

I

TABLE I

Ship ip.ad, knoto

expected circular frequency 8.53 0.893 10.82 0.955 13.10 0.987 15.43 1.037 0.697 0.600 0.555 0.515

1, dietance Of the "cefltre of

rotations to the FP, (metree) 83.3 81.1 82.5 82.0

E pitch angle,(radiaflS)2 0.00328 0.00393 0.00464

000464

pACC pitch motion at ACC, m 7.25 8.67 10.20 10.20

EPFP pitch otioU at FP, in' 16.05 19.25 22.70 22.70

heave, m2 1.19 1.33

234

2.51 E vertical acceleratiofl aALC 7.55 -. 1O;90 15.60 18.80 at ACC,

EACC vertical velocity at

9.5 11.9 22.6 18.0 39.3 31.4 12.0 13.2 25.8 23,5 45,7 41.5 16.0 16,4 31.6 30.7 53.0 51.6 17.3 16.2 31.2 33.5 57.0 61,.2 ACC,

(;)2

vertical motion sACC 2 at ACC, m vertical motion sFP at FP, m vertical velocity at Fp,

()4

irFP relatiVe vertical

motion at FP,

vrFP rilative vertical

velocity at _{FP, (--)}

E twice the variance

TABLE II

Frequency of. occUrrenCO of slamS, in per ceUt of the

number of relative motion oscillations

24

Slamming severity

Ship speed (knots)

8.53 10.82 13.10 15.43

> 0

g 19.5 35.0 47.5 52,5

> 0.1 g

9.5 .. 20.0 34.5 39.0 > 0.2 g 4.5 11.5 24.0 28.0

>

U.J g

4.0

6.5

16.5.

20.5

> 0.4 g 1.0 3.5 11.0 15.5

0.5 g

-

2,0

7.5

12.0

> 0.6 g

1.0

5.5 9.0

> 0.Tg

- - 4.0 7.0

> 08 g

. 5.5 > 0.9 g . - 1.5 4.0 > 1.0 g . - - 1.0 3.0

>l.1g

. -. - - 2.0

>i.2g

- -

1.5

> 1.3 g - - - 1.0

Note : the slam d.celeratiOnsref.r to station ACC

Valusi oE

a'

(.tr..)

25

TABLE III

S1aming .verity

Ship epeed (knote)

-8.53 10.82 13.10 15.43 0 g 8.05 6.95 6.25 6.05 > 0.1 g 9.60

860

7.50 7.30 > 0.2 g 11.10 9,95 8.7O 8.55 > 0.3 g 12.40 11.25 9.80 9.50 0.4 g 13.30 12.45 10.85 10.J0 > 0.5 g 13.45 11.75 11.00

> u.S

g - -. 14.7O 12.45 11.70 > 0.7 g - 13.10 12.35

>0.8 g

- - 14.O0 12.90 > 0.9 g - 14.95 13.60

>1.0 g

-

_T-

_{15.65 14.15}

>1.1 g

- _{- 14.95} >1..2 g - _{- 15.55}

Note : the ala. decelaratione tOfertá atation _ACC

26

Pi&iar.s

Lu-ship - Breadth of flat forebottom versus chip length

Ensrgyepectrum of irregular waves, significant height 6.8 a

Examples of Rayleigh distributions of relative motion

Histograms of slam d.oelerations at station ACC of "Luship"

in ballast condition

Cumulative density distributions of clam decelerations at tation

ACC of "Luchip" in ballast condition

0. Luship (ballast condition) - Probability of exceeding a value

of relative motion amplitude at the FP

7. Luship (ballast condition) - Relationship between squared "threshold velocities" and slamdeceleration levels at station

ACC fop different ship speeds

. Luchip (ballast condition) - ValueE of "threshold velocities" at

the FP versus chip speed, 'for different levels of clamming severity

9. Luship (ballast condition) - Calculated pitch angle amplitudes

in regular head waves of 1 metre amplitude

lu. luship (ballast coüdition) - Calculated heaving motion, amplitudes

in regular head waves of 1 metre amplitude

Luship (ballast condition) Calculated phase angles of pitching and heaving motions in regular head waves

(

Luship (ballast condition) - Calculated amplitudes of relative vertical motion at different stations in regular head waves of

1 metre amplitude for a ship speed of 10.82 knots

LUchip (ballast condition) - calculated amplitudes of relative vertical motion at different stations in regular head waves of

1 metre amplitude for a ship speed of 13.10 knota

Luship (ballast condition) - Decrease of relative vertical motions along the ship length in irregular head waves

Luship

(ballast condition) - Values of "threshold velocities"

station 0.1 L and 0.2 L aft of the FP versus ship spec4,

S1a deoslerations versus siu1tan.eus relative vertical Velocities

at the YP , ai measured on the model of the Lu-sbip (ballast

condition) in irregular wave8

Lu-ship (ballast condition) - Slam dec.lerations Verono simulta-neoUs relative Vertical velocities at the FP for a speed of

13.10 knots in irregular waves

Lu-ship (ballast cod.itOfl) - Slam.. decelerationa versus simultaneous

relative vertical velocities at the FP for' a speed of 15.43 knots in i:rregular waves

m.v. Jordaefls _{Relationship between slam deceleration and whipping}

stress in the main-deck at midship

U. Number of clams above a critical level per 100 otiOn oscillations for the Lu-shipm in a head sea 7 - .8 B.

16

12 B 4

B*(m)

15 10 5 1.6 .4 -, I I

---I

20t5

10 8 7 6 FIG. 2 5 T(sec) 4 0 FP .1L

,2L

FIG.1

20

-.3L

.4 L .5 L

.15 .10

.05

0 .2 .4

FIG. 3

FIG. 5 £

-f(sr):-e

sr

.6 .8 1.0 1.2

Ui -C-,

z

-C.,

0

0

0

>-C)

-z

Lii 0 Ui 20 U. 10 .4 . . ii? 11 .1 .2. V

10.82 knots

V: 1.5.43 knots

0 g ;47.2 per cent

FIG.4

Og:65.1 percent

2 3 4 5 6

91011

11.2 113 14

DECELERATION ('ACC') ,times

g

2.0 10 0 20 V

853

knots

Og:80.5 percent

0 .1 .4

V: 13.10 knots

Og:52.2 percent

.2

.6

.8

1.0 1.1

V: 8.53 knOts

10.82

13.10

15.43

0 9 -10 11 12 13 14. 15 16 1.7

RELATIVE MOTtON AMPLITUDE AT FP,

H' (metres)

2 1.0 .8 .6 .4 .2 0

modet tests: x V

10.82kn

o:V:13jO kn

FIG. 9

FIG. 10

V: 1310 knots

V:1O.82 knots

1.0 1.5 2.0

25

WAVE LENGTH/SHIP LENGTH A/L

z, HEAVE

PER WAVE AMPLITUDE AMPLITUDE

(m)

h :1 m

mOdeL tests

X:V:1Q82 kn

0 :V13..10 kh

x

.5 1.0 1.5 2.0

AlL

2.5

-150

-125

-100

- 75 - 50 - 25

PHASE ANGLES

(degrees

1.0 1.5 2.0

AlL

2.5 FLG. 11 HEAVE , PITCH ,

£

3 2 3 .5

tO

1.5 2.0

A/L

2.5.

FIG 12

FIG. 13 2.0

X/L 2.5

Cm) -.

V:1O.82kn

RELATIVE

(m)

MOTION AMPLITUDE

VERTItAL

.

-wave

.

ampLitude

.

:1 m

-4

V:13.i0

FIG 14

V:13iO kn

V;1O.S2kn

FP

.1L

i.oL.

.6 .4 IRREGUt.AR

SIGNIFICANT

HEAD WAVES:

HEIGHT: 6. m

0

--.1-0

4..

-Z

0

- a

Lii'

Jz

-4

-+ -4. ₀ 0

WI.-c4

1-(.DU) _J U) ₀ -x 0 -- -+ 0 + Ox 0 + X + - -

----. O X 0 + O

A

O 0 0 0

+ 0

4+

o _Ox 0

00

A

C4+

,( U 0 0 0 A -f X -0 0 0 x X0

0+

A 0 + 0 x 0 -1-u)Q + -fX

ttiX

+ O

XX +

+c-'4o

OODO U . oX

X00

)( _A

0-A

RELATIVE VER1TICAL + A +

VELOCITY AT FP

o 5 10

m.sec'

15 FIG 16

lii

1.0 1.2 .3 .2 I

V: 15.43 knots

0 0 0 FIG. 17 0 F I G .1 0 0 0 0

lii

0 0 0 OJ o0

/

00

0

00 /

₀ 0 ₉

,'

0 0 0

/

/

V

/0

F 0

_.1

0

/

0/

0 0

/

/

o 0 0 -0

/'

0 0 0 0 _co 0 0 0 0

070

I _I I I _I -I - I - I 3 4 5 6 7 8 9 10 11 12 13 14

RELATIVE VERTICAL VELOCITY AT FP (rnsec1)

/

/

0

V :3jQ knots

₀

/

A

V ₀

/.o

0 0

0/

0

000/

_,

₀ 0 0

0/

0₀ 0

0 70

00 0 06 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RELATIVE VERTICAL VELOCITY AT FP C rnseit)

.9 .8 In

.7_

E

0.

.6

<U

c-)

wz

-J L&Iz

3I-.2<

o

30 0 U)

0

25_

-J -J

0

cn

0

.

z

'0_0

LU- SHIP, BALLAST

HEAD SEA 718 B

0 .5 1.0 1.5

WHIPPING STRESS (kg/sq mm)

FIG. 19 10 12 14

SHIP SPEED V (knots)

FIG. 20

0

I

15 U)

I

4

-J U) LL

10 0

LU

I

z

0 0 0