Slamming on forced oscillating wedges at forward speed. Part II: Slamming simulation on penetrating wedges at forward speed

SLAMMING ON FORCED OSCILLTING WEDGES AT FORW2RD SPEED.

Part Ii: Slamming simulation on penetrating wedges at forward speed.

D. Radev* and W. Beukeiman** Report No. 890-P 1991 I.S.P., Volume 40, Nr.421, 1993

*

Bulgarian Ship Hydromechanics Centre., yama., .Bulagria, Research Fel.iow.at:Dè1ft University of Technology. **

Ship Hydromechanics Laboratory, Deift University Technology, The Netherlands.

DelftUnlversftyof Technology

Ship Hydromechanics Laboratory Mekelweg2

2628 Cl) Deift The Netherlands PhoneO15-786882

INTERNATIONAL SHIPBUILDING PROGRESS

Volume 40, noo 421

Press/1993-f

CO NTENTS

in Memoriam Professor Dr.-Ing. Helmut Völker

P.W. Ch'ng, L.J. Doctors, and. MR. Renilson,

'A Method of Càiculating the Ship-Bank Interaction Forces and

Moments in Restricted Water'

T. Svensson,

'The Significance of the Parameters of Some Closed-Form Expressions

for Estimating Cumulative Damage'

M.M. El-Gammal and M.A. Mósaad,

'Hydrodynamic and Technical Design Aspects of Surface Effèct Ship Technológy'

D. Radev and W Beukelman,

'Slamming on Forced Oscillating Wedges at Fòrward Speed'

Part H: Slamming Simulation on Penetrating Wedes

_{at Forward Speed 71-92}

X. Xinlian, J. Zhuoshang, and Y. Yn,

'Nonlineair Programming for Fleet 'Planning' _93-103

S

7-23

25-52

s

I IO ON FORCED OSCILLATING WEDGES AT FOR SPEED.

PJRT II.: SLAMMING SIMULATION ON PENETRATING WEDGES AT FORWARD SPEED.

D. Radev* and W. Beukelman**

*Buigarian _{Ship Hydrodynamics Centre,, Varna,,} Bulgaria.

Research Fellow at Ship Hydromechanics Lab Deif t UniversIty of Technology in 1991. The Netherlands.

**S.hip _{Hydromechanics Lab., Deift. University of} Technology, The Netherlands.

This paper presents an approach of sÏaimning in'-vestigation of 3-D penetrating wedges at forward speed using momentum theory. The research is a continuation of the method of Beukeirnan i98O

[3], including forward speed influence and 3-D hydrodynarnic effects.. The test results obtained by the Ship Hydromechanics Laboratory' of DeIft University of Technology [4] have made possible the dévelopment and implementation of this ex-panded method., A time simulation has been

per-formed for the wedge entering into the water. Computed results have been compared' with experi-mental data in such a way that the influence of the parameters considered. is clearly

Nomenclature

waterline beam of wedge. g acceleration of gravity

J. local reduction coefficient of added

mass along longitudinal direction L waterline length

m' sectional added mass N' sectional damping, p pressure s vertical displacement T draught t time U forward speed

V vertical Speed (upwards positive)

X01Y0.,Z0 right händ. coordinate system, fixed in space.

x,Y,z right hand) coordinate system, at. the water level moving with the wedge

speed.

XB,.YB,,ZB right hand coordinate system., fixed to

the wedge

half w:dth of the submerged cross section on the waterline

ß deadrise angi.e

p mass density of fluid

w circular frequency of oscillatIon

1. introduction

During the last few years, the efforts are being continued to de.termthe thé slamming pressure. mainly jn two directions:

- momentum' theory [2;, 9., 12;, 18., 21, 22, 23]

- impact theory. [2, 19, 20, 21., 22]..

MiyamotO., Tanizawa [.151 studied water impact

experimentally and numerically for wedges with small deadrise angles.. The 'authors' presented a mathematical model including air influence..

Kaplan [101 used: a quasi three dimensional

representation of the added. mass of the Section (without taking account of. frequency domain.) for advanced marine vehicles, on the basis of momen-tum theory.

Matusiak and Rantanen [13.]

proposed a un±fed

method for hydrodynamic loads. The linear por-tion of hydrodynamic loads was produced by the classical linear seakeeping strip theory. The:

'non- linear terms represent. primarily the intpact

loads called slamming. Added mass and damping of the sections as a function of draft are evalua-ted i.y a 2-D close-fit method (Bedel, Lee, .1971) Takemoto, Hashizumi., Oka [25'] measured wave

impact load and hull responses for a patrol boat, while the, boat was run in severe wavês at. different encounter frequencies and speeds.

Beukelman [31 presented a two. dimensional method on the basis of s.:trip:theory [7], using

Frank'-Close-Fit method for added mass calculation

[8]

The present research was performed with

pene-trating 3-D wedges at f orwárd speed in the most disputed zone: deadrise angles 00

30,

where the classical -D t1eory of Wagner [26] gives

infinite results for pressure, different from the test data of Chuang [5,, 24] and others [6]

(Fig.l). This report presents a continuation of the method of Beukelman [3], including forward

speed influence and 3-D hydroynamic effects

½pV2 [-1 300 p 200 io o _I

./

_, TBEORY - - - - . WAGNER EXPERIIIENTS CRUANG

I

YA}(AMOTO FlUITA O EAGIWARA 10- 20 B degrees

Figure 1. Variation of peak pressure with dead-rise angle for 2-D wedge shaped body

2.. Prediction method for determining the

slam-ming pressure on penetrating wedges at f or-ward speed.

The proposed calculation method is based on the momentum theory including forward speed [13.) and 3-D effects.

2.1.

i

I.'C

mode].

The coordinate systems relative to which the:

wedge j5 penetrated with vertical velocity V and f rwad speed U, are presented in Figure. 2. The right hand coordinate. system (X0.,Y0,Z0) is fixed in space. The Z0.-axis is vertically up-wards, the X0-axis is in the direction of the forward speed o.f the wedge. The coordinate system O(X,Y,Z) is moving. wi.th constant forward speed U. The. :(X,Y) -plane is situated in. 'the: s'ti].]. water surface., X in the direction of the wedge. speed U' and Z positive .upwards. The system

C(XB,YB,.ZB.), is fixed to the wedge, C being the

main corner point 'of the wedge.. The relation between différent coordinate systems is as follows:

X0 = Xc + XB cosa - ZB. ema

.,( I)

'go. = 'Zc + XB Sifla + ZB COSa,

where a is trim angle:, Xe,, Z are coordinates of the corner point C.

For the wedge, there 5 a linear correlation between _B and ZB:

ZB .= _B (2)

For pure heaving osc'illaticn of the: wedge with

forward. speed., the fo'ilowng appiies

Xc 'Ut.

Figure 2. Coordinate systems. Penetration of 3-D wedge in different times.

The hydrodynainic force per unit length acting. on each heaing section at position X is calculated by using Newman's f:ormuia

_{(1977) [13]}

d .. ., 1

F.(X,Z)= - - {s(x,z) [in (X,Z,)+ ---

N

(X,Z,w)]}-dt 1w

+ pgA

(4)

where: g - gravitational accele]ation

t - time

The operator - is thetotal derivative, with

dt

respect to time t,, defined as:

d 8 8

-=--

U-dt at ax

U - wedge forward speed

i - section heave in terms of the relative velocity.

m' sectional added mass

N' sectional damping,

- the difference between cross-sectional area in motion and in, still water

All the. above variables are function of time. Applying formula (5) for the derivative,

equa-tion (4) can be Written, as follows:

,. din'. dm'

F(X,Z). =-m.s-'Ns---,s2 +

Us'- -ds dN'. dN" + ,

SS + -

Us ds '(6)

a as a

where s Z for calm, water and - = -

= s

-at as at as

For slamming pressure, the following equation is obtained: i 8 ,.. dm '. dm' dN'. p(X,Z) =

- - (m s+Ns+

Us+ ss

-28Y

dS dS dN

+ - Us)

The slamming pressure achieves infinite value

f or flat bottom wedge (ß = O), or for the central point (Y = O) on the wedge.

Making comparison with the method of Beukelman [3] for the evaluation of the slamming pressure, the first three terms in the same. The fourth term is the influence of forward speed, including the change of added mass in X d1irec-tion [il]. The remaining terms show the mf lu-ence of damping derivatives, together with ver-ticàl and forward speed in fixed dIsplacement é.

2.2. Hydròdynamic

characteristics

Fôr calculation of the heave force load as well! as the peak pressure of the wedge entering the

water, it ïs necessary to know the added mass

and damping coefficients in the time domain. För typical 3-D. wedge forms, calculations are per-formed using the SHL method [1, 16] or Meyerhof f method for plates [14], for obtaining the local, reduction factor J(B/.L, X/L). This factor con-sists of the ratio between 2-D and 3-D coef f i-dent including general geometrIcal form and relative position of the cross section. along the longitudinal direct iono

2.3. Numerical realization

In the present study, the calculation of the slamming pressure is simply based on the time

derivation of the momentum of the added mass and damping coefficient and their derivatives. In

the time domain simulation, at every moment of penetration, there are always new wedges with the same initial global ratio of parameters, but with the change of the local position of the starting cross section X/L.

On the basis of few characteristic points for added mass and damping coefficients and their derivatives with respect to Z and X, the approx-imated curve is obtained for the duration of the simulation. This fitting is done by using

or-thogonal polynomials. The error obtained is too small, so that its influence for determination of the derivatives is neglected.

3. Numerical experiments

3.1. Geometrical parameters of test-wedges

In this research, the object of investigation

are 3-D wedges. The geometrical form of every wedge is determined by the following ratios:

T = L tga

B (8)

T = - tgß

The ratio between L and B is derived from the above two expressions:

L

ltgß

B - 2 tga

The main linear parameter is absent for wedges, because the global dimensions of the wedges are determined only by their proportions.

The global geometrical . ratios of ail the four test-wedges are shown in table 1.

Table 1. Geometrical characteristics of different 3-D wedges.

it is seen that the largest dif;ferences of cha-. racteristics for the wedges lie between a dead-.

rise angle less than 2 degrees and a deadrise

angle of 9.98 degrees. From the table., it is understood that the big influence is not only

from the deadrise angle, but also from the trim angle. The different variants, of these two angles give dtif:f.e.rent proportions of the main wedge parameters.

For the first,

second and third wedge with

deadrise angle. less than 2 degrees, the ratio L/B is very small:

L B

This . means. that it is not appropriate to.. apply

strip theory directly for the calculation of 2-D hydrody.namic coefficients. This is the reason

for applying 3-D calculation methods in this investigation. Ro. RATIO WEGE PARÑIETS L

-B L

-T B

-T 1.1 p - 0.090, - 0.50° 0.092 114.8 3.250 1.2 P - o.o°, - 2.50° 0.018 22.9; 120 2.1 p - 0.46°, o - 0.500 0.458 114.6 250 2.2 p - 0.46°, a 2.50° 0.092 22.9 250 3.1 p 1.15°. a - 0.500 1.146 114.6 100 3.2 p 1.150, « - 2.50° 0.229 229 ioo .1 _p 9.98°, a - 0.509 10.053 114.6 11.4 4.2 p - 9.98°, a - 2.50° 2.009 22.9 11.4

In f ig-ures 3 and 4 he chaiges of wetted area for the. first and the fourth wedge in ti,me do-main a.z'e shown. Big differences are so obsérved that the global ratio parameters, as well as X/L of these two wedges are changed during time do-main of penetration.

-3.2. Rydrodynmic coefficients

T1e hydrodyilarnic coefficients are calculated usin the Frank Close-Fit rtiethod and the Meyerhof f 3-D potential method for plates.

In f ig.5 the added mass per unit length calcula-ted by Frank Close-Fit method is presencalcula-ted.. it

is seen that f pr the same frequency the effect.

of changing the deadrise angle on added maSs coefficient is not sinifiant whereas for the

saine deadrise angle, the difference in added

mass coêfficient iS very high while changing the

f requei1y from O - 12 sfl-.

Similar tendencies are shown for damping coef f i-dents in f ig.6. The hydrödrnamic cöefficients .of the four test-wedges are shown in ;table 2.

zw

P - 0.09° ß - 0.46° p - 115° ß - 9.98°

S o' 9' S o' N' S o' N' S o' N'

t.] Io] No2

-

to] Itwoi to] 16821 I-1 1.2

(']

to] 18821 F-1 1o2 181] F-I F I 2 I 0.03120 0.00000 1.847 21.913 0.00025 1.830 21.882 0.00062 1.820 21.882 0.00550 1.600 21.700 0.06250 0.00010 6.134 64.766 0.00050 8.104 64.642 0.00125 6.076 64.396 0.01100 5.587 63.249 0.09375 0.00015 1.2.779 117.861 0.00075 12.710 117.560 0.00187 12.640 117.308 0.01650 11.660 112.000 0.125000.00020 21.869 177.110 0.00100 21.777 174.942 0.00250 21.61.5 174.461 0.02200 19.870 167.51'

Table Sectional hydrodynamic characteristics w 12 calculatéd by Prank

The graphs on f±gure 7 give the change of added mass in longitudinal direction of the wedge

(deadrise angle ß 9.980) _{in the t1me domain..}

The. mömentum added mass is Shown In time domain for a fixed strip relative to {XB.,YB,ZB} coor-dinate sys tern.

Fi'gure 8 shows the local reduction coefficient of the added mass calculated by the Mèyerhof.f 3-D potential method for plates concerning 2-3-D results. Large difference is observed for the

values of the added mass for different ratios of B/L and the distribution of X/L.

Figure 9 presents the added' mass calculations by strip theory and by the Meye]hof f 3- D potential method for the wedge with deadrise angle ß =

0.09° and trim, angle a = 0.5°. The big differen.-ce between these two' methods is there, where the penet rat ion increases,.

The. deviation of .addéd mass. during inItiai stage

of wedge penetration is not so large, but this results in a significant influence on the. value of the derivatives.

X X 5 0 090 Ja O.S0' J I .Ó048j 'X X z y

Figure 3. Changes of wetted area for 3-D wedge in tinte doain.

I

t

o j B 9.980 2.5 j ZB.528 C' z c.) C'

Figure 4. _{Changes of wetted area for 3-D wedge}

20 10--0-- 8 0.090. w- O .I - 00.09°. .12 . ®__ 09.9O. 12

-

0.12 0 -00.09. .12 O 12 . 0.02200 0.01100 I

Figure 5. Added mass per.unit length for

dif-ferent 2-D wedges calculated by. Frank

close-f it method.

Figure 6. Damping coefficients per unit length for different 2D wedges calculated by Frank close-fit method.

0.00010 0. 00020 0.01100 0. 02200 200 t,. 100 0.06 2w (io]

*-

0.12 0.00010 -w- 0. 0 0020

0.5 o 9.98e a - 2.5e

--t-o

O C(Xj1,Y21,Z20.0018) I e e o -S -rn -S--S -S-1.0 o 5 X8 L XB_______ L

Figure 7.. Distribution of added mass in longi-tudinal direction of wedge in time domain.

-D/L

1.0

Fi,gure 8. Local reduction coefficient of added mass along longitudinal direction for plates. (W. Meyerhof f, .97O)

.10

0.5

J(X,pc)

'MIII",

2

n

- 10.9. 15/L0.27

- - - STEIP 2050ES - YERE0YP FOR PLATES

7

0.06

0.00010

0.12

0.00020

Figure 9

Added mass by strip theory and by

Meyerhof f 3-D method for plates.

4. COMPARISON WiTh TEST RESULTS

The figures 10 - 17 show the Slanning pressures

calculated by the present method in comparison with tests results [4]. Every figure gives

in-formation regarding geometrical characteristics of the wedge (deadrise angle and trim angle) and kinematic conditiönS. Also the distribution of the points of investigation is shown.

Graphs 10, 11, 12 and 13 give thé comparison between test points and predicted results of the peak pressure in time domain The correction of added mass is determined by using Meyerhoff's 3-D method. The result is rather high for

calcula-tion With 2-D coefficients.

From f igues 10 and 11 good correlation for

deadrise angle ß 0.090 is seen in the case of change of vertical speed.

Where forward speed is changed, the deviation from the test points is higher. When comparing the peak pressure in time. domain for different wedges (ß _{0.09°, ß = 9.98°) the rapid change}

in the peak pressure f.or wedges with small dead-rise angle 5 _{clearly shown.}

in figures 14, .15 and 16 test points, and pre-di.cted results of the peak pressure: are shown as a f.únct.ion of vertical speed for two. forward speeds. The. difference in predicted results for changing forward speed is too emáli. The expia.-nation is that. the term which include forward

speed in formula ('7) has small influence for

flat wedges

(a -> .0°').

Figure 17 'presents' t'est point's and predicted results of peak pressures as a function of forward' speed.. For V = 0.24. rn/s the results are better then' those for V = 0.72 rn/s..

Based on t'he analysis from the obtained, results, the following remarks' can be made. with respect to the method presented' in the report:

the term proportional to the squared ver-tical strip speed including derivative of added mass' has the i'arges.t influence,

specially in the case o.f flat wedges (ß ->0e);

- the last two terms in formula ('7), which inc'lude the displacement and damping coefficient have small cöntribution for flat wedges (ß -> 0°), because displace

ment is small (s -> 0);

the influence of the term which consist's of forward speed and derivative 'of' added' mass in X direction is higher for wedges with greater trim angles.

p

(kpa]

30-20

.10

-

o----13=0.090 a =

0.50

3.,OOm/s W O.72m/s __fU 3,.00rn/s O _{- O.24m/s} 0 0.022 0.. 044

t (lusec]

Figure 10,.. Test points

( o ). and predicted

results _{( -, - - ) of peak pressure} in time domain.

o 0.022 0.044

t [nisec]

Figure 11. Test points ₍ s o ) and: predicted

results

( -, -

_{- ) of peak pressure} in time domain.

o _o." 0.22

t [msec]

Figure 12. Test

points

₍

.

_o

)

and

predicted results _{( -, - - ) of peak pressure} in time domain.

30

f

20, p [kPa,] lo o 9. 98

o AØ

2.. 50 -.- ¡Ú 3.00mf2 IV = fU 1.0Dm/a o 3.3 6.6 t (insec

Figure 13. Test points _{( .} o ) and predicted

results _{( -, - - ) of peak pressure} in tinte domain.

t

p

[kPa]

Figure 14. 'Test, points'

( s o ) and predicted

results (s., o--) _{of peak pressure} as function of vertical velocity V.

V [m/s]

Figure: 15. Test points

( . o ). and. predicted results _{(-.-, -) of:; peak pressure} as function of vertical, velocity _V.

0 0.24 0.48 0.72

V (Th/s]

Figure 16. Test points ₍ . o ) and prediöted

results (s--, o) of peak

pressure

p

[kPa]

o .1 _{2 3}

U []fl/:S]

Figure 17. Test points

( . o ) and predicted

results (., o--)

of peak pressure

5. Conclusions and recommendations

On the basis of performed investigation,

the

following conclusions and reconunendations can be

drawn:

i. the proposed expanded method is hereby

sug-gested for evaluation of the peak pressure

including 3-D effects and forward speed

in-fluence. A time simulation program on

pene-trating wedges at forward speed is realized;

while considering 3-D effects (including the

influence, of the global geometrical

parame-ters and load position of the cross section),

ealuation of the peak. pressure for fiat

wedges gives relatively good agreement with

test results,;

the influence of the term, which represents

the effect of forward speed is increased for

wedges with larger trim angles;

the term proportional to the vertical strip

speed squared multiplied by derivative of

added mass has the most dominant influence

for evaluation of peak préssure';

5, further investigations require evaluation

of

hydrodynamïc coefficients on the basis of

advanced 3-D methods, as well as free surface

effects;

6. the experience from' such slamming, simulation

on penetrating wedges at forward speed can be

applied for ship slanuning.

This investigation is carried out in the Ship Hydromechanics Laboratory of De] ft University of Technology.. The first part including test

results is realized by ing. W. Beukelman. The prediction, method is developed by Dr. D. Radev, during his fellowship in Deif t. The authors would like to express their sincere gratitude to the department staff of Ship Hydromechanics Laboratory for undertaking such noble research.

7. References

1] Adegeest, L.J.M.,

'Analysis of a Three-Dimensional Method for the' Calculation of the Distribution of Hy-drodynainic Coefficients and Ship Motions with Forward Speed', .Deif t University of Technology, SRL, Report No. 845-S, 19:89.

6. Aknowledgmnt

( 51 Chuang, S

'Slammng Models. in Report No.

Bishop,. R.,E..DI and Price, W., of Bottom, and Flare Slamming on Responses", Trans. RINA .1987.

[ 3] Beuke]!man., W.,

'Bottom impact Pzessures due tO Forced Osc±llation, ISP Volume 27, 195:0.

4] Beukelman, W..,

'Slarrn.ing on. .f.orced'oscill'ation wedges at forward speed, part i: test. results', DeIf t University of Technology,, Ship Hydronecha-nics Láboratory, Report Nb. 888, 1991,, ISP, Volume 1991.

Tests of Three:- Dimensional Calm Water and Wàves', NSRDC 4095, 1973.

[ 2], Belik, O.,

' Influence Structural

6] Faitinsen,, 'O.,

'Sea Loads on Ships and Offshore Struc-tures', Camb,ridge University Press, 1990.

'L 7] Gerritsma, J. and Beukelman, W.,

'Analysis of' the Modified Strip Theory fr the calculation of Ship Motions and Wave

Bending Moments, ISP', Volume 14, 19:67.

[ 8] Frank, W, and Salvesen, N.,

The Fran'k Close - Fit Motion' Computer Program", NSRDC Report No.3289, 1970.

{ 9'] Jensen, J. and Pedersen, P.T.,

'Wave- induced Bending 'Moments in Ship- A Quad:ratic Theory', Trans. RINA, 1978. [10'] Kaplan, P.,

'Analysis and Prediction of Flat Bottom Slanning Impact of Advanced Marine Vehiçles

,in Waves', ISP Vol.34, 1987".

[li] Keuning, J.,

'Distribution of Added Mass 'and Damp±ng Along t'he length of a Ship Model Moving at High Forward' Speed', Deif t University of Technology, Ship Hydromechanics Laboratory, Report No,,817'-p, 19:88 and ISP vol.37, 1990.

Mansour', A. and d'O]Liveira, J.,

'Hull Bending Moment Due to Ship Bottom Slamming in Regular Waves", JSR Volume 19,,

1975.,

Matusiak, J. and Ratanen,, A.,

'Digital Simulation of the Non-Line:ar Wave Loads' and' Response of a Non-Ridgid Ship', CADMO 19:8:6, Washington D.C.

Meyerhoff1 W.,

"Added masses of Thin Rectangular Plates Calculated from Potential Theory", JSR,, Volume 15, 1970.

[15] Miyamoto, T. and Tanizawa, K.,

'A Study of the 'impact' Load on Ship Bow', 2nd Report, Journal ,f the Society of Naval Architects of Japan, Volume 158, 1985.

[16,] Mynett,, A and Keuning, J.,

'Ocean Wave. Pata Analyses and Ship

Dynamics', Memoriál Symposium to Professor' R..E.D. Bishop on the Dynamics of Marine Vehicles and Sttuctures. in Waves, Brunell University, 24 27 June 19.90, London., U.K.,

Ochi, M..K. and Motter, L.,

Prediction, of Slamming Characteristics and

Hull Responses'. för Ship. Design', SNAI 81,

1973.

Petersen, J., and Marnts,, L.,

'Comparison of Non-Linear Strip Theory Pre-dictions and Model Experiments',, PRADS '89, Varna.,, Bulgaria.

['19] Radëv, D.,

'Numerical Realization of' pröblem for Entry' of Rigid Circular Cylinder in Compressible Fluid', IV th .IMAEM Congress 1987, yama.. [20] Radev, D.,

'General Approach to Ship Bottom Slamming Investigation' IVth IMAEM congress 1987,

yama.

[21], Report of' Cöimni'ttêe 11.2 "Dynamic Load Effects'!,, 10th ISS'C 1988, Denmark.

Report of the ITTC Seakeepi.ng' Committee., Proc. of the. 19th ITTC, 1990, Madrid. Soares, C.,

'Transient Response of Ship Hulls to Wave

[24] Stavovy, A. and Chuang, S..,

'Analitical Determination of Slamming Pres-sures for High-Speed Vehicles in Waves',, JSR 1976.

[251 Takemoto,, H., Hashizumi,, Y. and Oka, S.,,

'FuliScaie Measurement of Wave Impact Loads and Hull Responses of a Ship in

Waves', Ist' Report, Journal of the Soc. of Naval Architects of Japan,, Vol. 158, 1985. [261 Wagner, V.,

'tJbers;tos.z- und gleitvorgânge an der Ober-f láche von Fiûss:igkeiten' , Zéitschrift f Ür

Angewandte Màthematik und Mechanik, Vol.12, No.4, 1932.

SLAMMING ON FORCED OSCILLATING WEDGES AT FORWARD SPEED.

PART II,: SLA}!MIÑG SIMULATION ON

PENETRATING WEDGES. AT FORWARD SPEED. D. Radev * Report No.890 Aprii 199.1 *

Bulgarian Ship Hydrodynamics _{Centre, Varna,}

Research Fellow at Deift University _{of Technology.}

-DeiftUfliverelty of Technology

Ship Hydromechanics Laboratory M8kelweg 2

2628 CD Deift The Netherlands Phone 015- 786882

General information

This

investigation

_{is realized by Dr. D. Radev from Bulgarian}

Ship Hydrodynamics Centre during _{the period of his research} fellowship at Deift University of _{Technology, Ship} Hydrome-chanics Laboratory.

The document is to be considerad together with Report No.888 by W. Beukelman - "Slamming on forced oscillating wedges at forward speed", Part I: test results [4].

Photographs of the test-wedges together with researchers Ing. W. Beukelman (on the right) [4] and Dr. D. Radev (on the left).

CONTENTS

ABSTRACT

_i

NOMENCLATURE

2

INTRODUCTION ₃

PREDICTION METHOD FOR:,: DETERMINING:

_{THE:SLAING PRESSURE ON}

:,PENETRATINGWEDGES .,AT:FORWARD.SPEED..,.,. . 5 2.. 1.. ;Dynamic .mode].... . ..

. ...,.,.

... .

_{. . . .}

5

.:2.2.. Hydrodynainic. characteristics

.

...

₈

2 3 .

_{Numerical, realization...}

,,

. ..., , , . ,. .

.. . -. . ..- . .. . . ..

. . ₈ NUMERICAL EXPERIMENTS.

.. . .

. . .

₁₀

3.1. Geometrica], parameters of, test

wedges

10

3.2. Hydrodynainic coefficients

. 11

COMPARISON WITH TEST RESULTS ₁₈

CONCLUSiONS AND. RECOENDATIONS

.. ..

₂₄

ACNOWLE DGEMENT _{. s s s s s s .,s s s - e....-. :5 . .0 s. s: s . s s . rs s s s. s s s s ss. s s s i s s}

2 5

REFERENCES _{5 05 S . s ... s s s,.. s . s -..-s 2:6}

Appendix 1: Input file of slamming.

_{simulation program}

₂₉

ABSTRACT

This report presents an approach of slamming investigation of 3-D penetrating wedges at forward speed using momentum theory. The research is a

continuation

_{of the method of}

BeukeIman 1980 [3], including forward _{speed influence, and 3-D} hydrodynamic effects. The test _{results obtained by the Ship} Hydromechanics Laboratory (SHL) of Delft University of

Technology [4] hase made possible the development and implementation of this expanded _{method. A. time simulation}

has been .performed for.i...the.; wedge..entering..:jnto _{the. water.}

:..:.J..[CoIuputed results-have beencompared _{withexperimental data- j} such a. way _{that ..the .inf luence.of.theparameterscong}

idered .is clearly demonstrated.

NOMENCLATURE

waterline beam of wedge

dimensional (e.g. 3-D - three

dimensional)

acceleration of gravity

local reduction coefficient of addéd mass along longitudinal direction

waterline length

;:gectjonai added mass

sectional damping pressure vertical d'isplacement draught time forward speed

vertical speed (upwards positive)

X01Y,, Z0 _{right. hand.coordnate system, fixed in, space}

X,Y,Z _{right hand coordinate system, at. the water level} moving with the wedge speed

XB,YB,ZB.

right hand'.coordinate.gygtem,;.fjxed _tó.thewedge

hai f. widthofthe submerged3crossysec _{ion onthe} waterline

a _{trim angie}

ß d'eadrise'angle

p _{mass density of fiuid}

circular frequency of oscillation

2 B D

g

J

L In N'

p

g T

t

U V

1. INTRODUCTION

During the last few years, the efforts are being continued to determine the slamming _{pressure mainly in two directions:}

- momentum theory [2, 9, 10, 18, 21, 22, 23],

- impact theory [2, 19, 20, 21, 22].

Miyamoto, Tanizawa (15] studied water impact experimentally and numerically for.wedges,with.smal]. _{deadrise angles. The} authors presented a:. mathematical _{model including air}

influence.

Kaplan [11] used.a _{quasi'.three..,dimensiona1.represetatj0 of}

the added mass of the section _{(without taking account of}

frequency domain) for advanced marine vehicles, on the basis of momentum theory.

Matusiak and Rantanen [13] _{proposed a unified method for}

hydrodynamjc loads. The linear portion of hydrodynamic loads was produced by the classical linear seakeeping strip theory. The non-linear terms _{represent primarily the impact} loads called slamming. Added mass and damping of the sections

as a function of draft are evaluated _{by a 2-D close-fit}

method (Bedel, Lee, 1971).

Takemoto, Hashizumi, Oka [25] _{measured wave impact load and} hull responses for a patrol boat, while the boat was run in

severe waves at different encounter _{frequencies and speeds.}

Beukelman [3] presented a two dimensional method on the basis of strip-theory [7], using _{Frank-Close-Fit method for added} mass calculation [8].

The _{present research was performed with}

penetrating 3-D wedges at forward speed in the _{most disputed zone: deadrise} angles 00 - 30, where the classical _{2-D theory of Wagner [26]}

gives infinite results for pressure, different from the test data of Chuang [5, _{24) and others [6] (Fig.l). This report}

presents a _{Continuation of the method of Beukelman}

[3], including forward Speed influence and 3-D _hydrodynamjc effects.. 30:0 p 200

¼pV2

E-]

loo

s o THEORY

- - - - WAGNER

EXPERIMENTS CRUANC

I

FUJITA O HAGIWARA

-4

10 20

Figure 1. Variation of peak pressure with dead-rise angle for 2-D wedge shaped body.

2. PREDICTION METhOD FOR DETERMINING THE SL14MING PRESSURE ON PENETRATING WEDGES AT FORWARD SPEED.

The proposed calculation method is based _{on the momentum}

theory including forward speed _{(13] and 3-D effects.}

2.1. Dynamic model

The coordinate systems relative to which the wedge is penetrated with vertical velocity V and forward speed U are presented in Fig.2. The right, hand s coordinate system (X0,101Z0) is fixed in space. The Z0-axis is vertically upwards, the X0-axis is in the direction _{of. the. forward speed} of the wedge. The coordinate _{system O(X,Y,Z) .'is'.moving.with} constant forward speed U. The (X,Y)-plane is situated in the still water surface, X in the direction of the wedge speed U and Z positive upwards. The system _{C(XB,YB,ZB) is fixed to} the wedge, C being the main _{corner point of the wedge.}

The _{relation between different coordinate}

systems is as follows:

XoXc+XBcosa_zBsina

Zo = Zc + XB Sino + ZB cosa, (1)

where a is trim angle, X, _{Zc are, coordinates of.the.corner} point C.

For the wedge, there is _{a linear correlation between}

B and

ZB:

ZB = _{'B (2)}

For pure heaving oscillation of the _{wedge with forward speed,} the following applies:

Xc Ut

Z = _{= 5a coswt} (3)

The hydrodynamjc force per unit length acting on each heaving

-5-t

t

section at position X is calculated by using Newman's formula (1977) [13]: d . , 1 - - {s.(X,Z)[m

_(X,Z,w)+

_{--- N}

_{(X,Z,w)]) -pgtA}

₍₄₎

dt _1w

where: _{g - gravitational acceleration} t - time

d

The operator - is the, total derivative with respect to time

t, defined as:

da

a

dt at Ç5)

U - wedge forward speed

s - section heave in terms of the relative _velocity - sectional added mass

N' - sectional damping

- the difference between cross-sectional _{area in} motion and in stili water.

Ali the above variables _{are function 'of time.}

Applying formula (5) _{for the:derivative, .equation.:..(4) can be} written as follows: I I.. . dm dm F'(X,Z) = - ni s - N s 2 _{+ Us} ds _dXB dN' . dN' Us' ds _dXB a a as . a

where s = Z for' cairn water and

- = - -

_{s -.}

at as at as

For slamming pressure, the following equation _{is obtained:}

i ô .. dm'

. dm' dN' dN'

p(X,Z) = - - (iu s+Ns+

_{- s2}

_Us+

_{ss - Us)}

2 ÖY dS _dXB dS _dXB

The slamming pressure achieves infinite value for flat bottom wedge (ß = O), or for the central point _{(Y = O) on the wedge.} Making comparison with the _{method of Beukeiman [3] for the}

evaluation of the slamming pressure, the first three terms in (7) are the same. The fourth term is _{the influence of forward} speed, _{including the change of added mass in X direction}

[12]. The remaining terms _{show the influence of damping} derivatives, _{together with vertical: and forward speed in}

fixed displacement s.

2.2. Hydrodynamic characteristics

For calculation of the heave force load as well as the peak pressure of the wedge entering the water, it is necessary to

know the added mass and damping _{coefficients in the time} domain. For typical 3-D wedge forms, calculations are

performed using the SHL method [1, _{16] or Meyerhof f method} for plates [14], _{for obtaining the local reduction factor} J(B/L, X/L). This factor consists _{of the ratio between 2-D} and 3-D coefficient including.,.general _{geometrical form and} relative position of the cross section _{along the longitudinal} direction.

2.3. Numerical realization

In the present study, the calculation of the slamming pressure is _{simply based on the time derivation of the}

momentum of the added mass and damping coefficient and their derivatives. In the time domain simulation, _{at every moment} of penetration, there _{are always new wedges with the same} initial global ratio of parameters, but with the change of

-8

the local position of the starting _{cross section X/L.}

On the basis of few characteristic points for added mass and damping coefficients and their derivatives with respect to Z and X, the approximated curve is obtained for the duration of the simulation.

This fitting is done by usng orthogonal

polynomials. The error obtained is too small, so that its

3. NUMERICAL EXPERIMENTS

3 1. Geometrical parameters of test-wedges

In this research, the object of investigation _{are 3-D wedges.}

The geometrical form of _{every wedge is determined by the} following ratios:

T =Ltga

B (8)

T =tgß

The. ratio, between Land B is..;.derived»'_{from 'the :.:above 'two} expressions:

Litgß

,

B 2 tgcz

The main linear parameter 'is absent _{.,for wedges, 'because the} global dimensions of the wedges _{are determined only by their} proport ions..

The global geometrical ratios of all the four test-wedges _are shown in table 1.

Table i. Geometrical _{çh'aracteristjcso.f different 3-D}

wedges.. 10 -No. WEDGE

-,

PARAMETERS RATIO ' . L

-

-

L B L

-'

T B/. B T 5/2 ' T 1.1 ß = 0.09°, 0.50° '.0.183 L 0.092 114.6 625 1250 1.2 ß= 0.09°, 2.50° 0.037, 0.L018 22.9 625 '1250 2.1 ß 0.46°, = 0.50° 0917 0.458 114.6 .125 250L 2.2 ß 0.46°, 2.50° 0.183 0.092 ' 22'.g 125 250 3..1 , ß 1.15°, 0.50° 2.292 1l46 114.6 so loo 3.2 L 1.15°, 2.50° ' 0.458 O.229 22.9 50 100 4.1 ß - 9.98°, 0.50° 20.105L10.053 114.6, 5.7 11.4 4.2 _ß 998°, 2.50° 4.018' 2.009 22.9 5.7 ., 11.4

It is seen that the largest differences _{of characteristics} for the wedges lie between _{a deadrise angle less than 2}

degrees and a deadrise angle of _{9.98 degrees. From the table,} it is understood that the big influence is not only from the deadrise angle,, but also from the trim angle. The different variants of these two angles give a different proportions of the main wedge parameters.

For the first, _{second, and third wedge with deadrise angle}

less than 2 degrees, the ratio L/B is _{very small:} L

- <1

B

:.Thig,means

.directiy:.;..-for»...the

cients. This is the;- reason

methods in this investigation..

in 'figures 3 and 4 'the changes _{of wetted 'area for the first} and the fourth wedge. in time domain are shown. Big differen-ces are so -observed that the, global ratio _{parameters, as well} as X/L of these two wedges are changed during _{time domain of,} penetration.

3.2. Hydrodynamic coefficients

The hydrodynamic; coef.ficientsareca'icu1ated'1_{using.: the'."Frank}

Close-Fit method: _for

plates.

In f ig.5 the.. added mass.per unit _{length calculated byFrank} Close-Fit method is presented. It is. seen that for the same frequency the effect of changing the deadrise _{angle on added}

mass coefficient is not significant whereas _{for the same} deadrise angle., the. difference in added _{mass coefficient is} very high while changing the frequency from

_{O - 12 s.}

figure 6.. _{The hydrodynamic coefficients of the four} test-wedges are shown in table 2.

Table 2.. Sectional

_hydrodynamic

_{characteristics w = 12 S} calct:iated by Frank Close-Fit method,.

The graphs on figure 7 give the

_{change of added mass in}

longitudinal direction of the wedge _{(deadrise angle ß} = 9.980) in the time _{domain. The momentum added mass Is shown}

in time domain for a fixed strip

_{relative to (XB,YB,ZB)}

coordinate system.

Figure 8 shows the local reductioncoeffjcjent,of. the added mass . calculated by the Neyerhoff 3D .potertial mthòd for plates concerning 2-D results.. Large differences. js:.Øbseed for the values of the added, _{mass for different}

. ratios of B/L and the distribution of X/L.

Figure 9 presents the added mass calculations by strip theory and by the Meyerhof f 3-D 'potential met_{hod for the wedge with} deadrise angle ß = 0.09° and trim aigle

a =

0.50.

The big difference between these two methods is there, where the penetration increases. The deviation of added mass during initiai stage of wedge penetration _{is not so large,, but this} results in a significant Influence

_{on the value of the}

derivatives.

12

-ßO.O99 _ßO.46°

_ß1.i5°

_ß9.98°

S H rn' N' S m' N' S m' N' S rn' N' (M) (M) [NS2

I

LM2 1M2

i

i (M) INS21

II

LM2 j ÌNSJ, LM INs2l I-1 LM2J [CNSJ

I

LM (M] [NS2I

I]

. LM2] [(NS)

I-LM2 I LM2 Ò.03125 0.000051 1.847 21:913 o.000s 1.830 : 21.882 0.00062 1.820. 21.882 .00sso 1.600 21.700 0.06250l 0.000101 6.134 64.766: 0.00050 6.104 64.16421 0.00125 607,6 64:.396! O01100 5.587 63.249 0.093751 0.00015! 12.779. 117.861: 0.00075 12.710 _{117.560 0100187 12.640! 117.308! 0.01650 11.660 112.000} 0.12500 0.00020 21.869 177.1101 0.00100 21.777 .1174.942! 0.00250 21.615 _{174461 002200 19670. 167.514}

t

o X 3 0.09e

a0.50

Jc

(x1,

ZB .0048: y t = tu X z G1

t=

o z o cl X

t = ti

X

Figure 3. Changes of wetted area for 3-D wedge Figure 4. Changes of wetted area för 3-D wedge

in time domain. in time domain.

X

vi

B 9.98e I a

c(x1,

Y=3, Z8=

200

i,

[Ns/Ms] 100 o 14

-Figure 5. Added mass per unit length for different 2-D wedges calculated by Frank close-fit method.

-ß 0.O9. ù1Z s

® ._._...99gO, w2 s

Figure 6. Damping coefflcients per unit length for different 2-D wedges calculated by Frank close-fit method.

0 .00 010 [M) 0.00020

0.5

05

XB L

1.0 j0.5i J(X,,c) o O

0.5

1.0

XB L

Figure 8. Local reduc.tion,:cöef;ficient of..addëdmass along. longitudinal .dir.ec.tion.for plates.

(W. .Meyerhoff 1970) K = B/L 10

16

-, .10 K 2.0

_

rn' 20 = 0.09e

a050

J 10.9, XB/L0.27

-

STRIP THEORY

- I4EYERHOFF FOR PLATES

7

7

'wEMl

7

7

S[M]

Figure 9. Added mass by strip theory and by Meyerhof f 3-D method for plates.

7,

7

7

7

0.06 0.12

o.

ooòio _0.00020

4. CONPRISON WITH TEST RESULTS

The figures 10 - 17 show the slamming pressures calculated by

the present method in comparison with _{tests results [4].}

Every figure gives information regarding geometrical charac-teristics of the wedge (deadrise _{angle and trim angle) and} kinematic conditions. Also the distribution of the points of investigation is shown.

Graphs 10, il, _{12 and 13 give the comparison between test} points and predicted results of _{the peak..pressure in time}

domain. The . correction, of. added mass is .determined

by using Meyerhoff's 3-D method. The result. is' _{rather: higirforcalcu...} lation with 2-D coefficients. _{From figures. .10 and 11: good} correlation for deadrise angle

ß = 0.090

_{is seen inthe case} of change of vertical: speed. Where forward speed is changed, the deviation from the test points is higher. When comparing

the peak pressure in time domain _{for different wedges}

_{(ß =}

0.09°, _{ß = 9.98e) the rapid change in the peak pressure for}

wedges with small deadrise angle is _{clearly shown.}

In figures 14, 15 and 16 test points and predicted results of the peak pressure are shown as a function of vertical speed for two forward speeds. The difference _{in predicted results} for changing forward speed.is too _{small. The explanation is} that the term which include forward _{speed in formula"(7) has}

small influence for flat wedges

(a -> 00).

Figure 17 presents, test points _{and predicted results of peak} pressures as a function of forward speed. _{For V = 0.24 H/S} the results are better then those _{for V = 0.72 H/S.}

Based on the analysis from the obtained results, the following remarks can be made with _{respect to the method} presented in the report:

- the term proportional to the squared vertical strip speed _{including derivative of added mass has the} largest influence, _{specially in the case of flat} wedges (ß -> 0e);

-- the last two ternis in formula (7), which include the displacement and damping coefficient have small contribution för flat wedges _(ß

->

00),

_because displacement is small (s -> 0');

- the influence of the term which consists of. forward speed and derivative of added _{mass in X direction. is} higher for wedges with greater _{trim angles..}

p [kpa 30 20 10

o-ß=O.O9 a = 0.5e ru3.00HIS

-

lv= 0.72)1/S FU=3.00M/S

0v ..

0. 24 H/S p 30 20

10-N

N

N

J =ØØ90 cx = 0.50 f U-3.00H/S lv 0. 72 H/S r U=i.00MIS 0. 72 H/S 0.022 0.044 0 0.022 0.044 t {MSEC}

_t

[MSECJ

Figure 10. Test points ( O) and predicted Figure 11. Test points ( O) andpredicted

results C-, --) of peak pressure results (-, --) of peak pressure

in time domain. in time domain.

Figure 12. Test points (S O ) and predicted results (-, --) of peak pressure in time domain.

t [MsEc]

Figure 13. Test points C. O ) and predicted results (-, --) of peak pressure in time domain.

30

10

o

v[M/sJ

Figure 14.. Test points C O ) and predicted

results (-O-, O-) of peak pressure as function of vertical velocity V.

30

20

10

o

V _[MISI

Figure 15. Test points C

O S )

and predicted

results (O-,

O)

of peak pressure as function of vertical velocity V.

t

20

p

k?a)

I L

[kPa

y {NIsJ

-Figure 16. Test pöints

(O O)

and predicted

results (o,

O)

of peak pressure

as function of vertical velocity V.

p (kPa 30 20 lo

ß=9.980

O

.0)V

a= 2.50

O -G-Va 0.24 H/S .-.-V0.72 fl/S

V

u [H/s]

.

O 1 2 3

Figure 17. Test points (o e ) and predicted results (-O, ---) of peak pressure as function of forward speed ¡J.

5. CONCLUSIONS AND RECOMMENDATIONS

On the basis of performed

investigation,

the following conclusions and recommendations _{can be drawn:}

- the proposed expai,ded method _{s hereby suggested for} evaluation of the peak pressure including 3-D effects and forward speed influence. A tinte simulation program on penetrating wedges at forward speed is realized;;

..while;.considering3 -Deffectsf; ncludingthe- influence

of

y

test resu1ts;

-the infiuence.of:::the..term,:whjch.repregeflts _{the effect}

of forward speed is increased for wedges with larger trim angles;

- the term proportional to the vertical strip _speed

squared muItiplied by derIvative of added mass has the most dominant influence for evaluation of peak

pressure;

further .investigations _{require.evaluation ; of}

.,hydrodynamic:coeffic:ientsonthebasisof(.,advanced3D methods,, as well 'as'.free:'surfaceeffectg;

-".;

- the «experience _{'from 'such slamming simulation on} penetrating wedges at forward speed can be applied for ship slamming.

-AKNOWLEDGEMENT

The author would like to express his sincere gratitude to the head of Shiphydroiuechanjcs _{Laboratory Prof. Dr. Ir. J.A.} Pinkster for inviting him to _{carry out this research project.} Specially, extremely grateful to Mr. Wirn Beukelman, who has relied on the author's abilities, in connection with such an Interesting subject for investigation and for supporting the author's activities during the period _{of this research work.}

.rhanksarealso to. Mr,. _and

drawing thepictures.

Finally:,. the author

f

26 -REFERENCES

Adegeest, L.J.M.,

'Analysis of a Three-Dimensional Method _{for the} Calcu1aton of the Distribution of Hydrodynamic

Coefficients and Ship Motions with Forward _{Speed', Deift} University of Technology, SHL, Report _{No. 845-S, 1989.}

Belik, O., Bishop, R.E.D. and Price,, _W.,

:,.influence of Bottomand.F1are'siammi.ng

_onStrctral.

Responses', Trans. RINA 1987.

i: 3] Beukelman,, W.,

Bottom Impact Volume 27, 1980.

BeukeIman, W.,

'Slamming on forced oscillation wedges at forward speed, part I.: _{test results'., Delft University of technology,} SHL, Report No.888, 1991.,

Chuang, S.,

'Slamming 'Testsof Three-Dimensional.:Models in Calm Water and Waves', NSRDCReportNo.40.95, _1973.

[ 6] Faitinsen, O.,

!Sea. Loads on. .Sh'ipsand;Offghore,structu.res_',Cambridge University Press, 1990.

E 7] Gerritama, J. and Beukelman, W.,

'Analysis of the Modified Strip Theory for the

calcula-t'ion _{of Ship 'Motions and Wave Bending}

Moments, ISP Volume 1.4, 1967.

C 8] Frank, W. and Saivesen., N.,

'The Frank Close-Fit Motion _{Computer Program', NSRDC} Report No.3289, 1970.

[15]. Miyamoto, T. and 'A Study of the. Journal of, the. Volume 158, 1985

.Tanizawa,., K.,

Impact. Load.on;..Ship 'Bow'..,'. .;2nd.Report,

Society... of:Naval::.Archjtects'...o:f Japan,

Mynett, A. and Keuning, J.,

'Ocean Wave Data Analyses and Ship Dynamics', Memorial

Symposium to Professor R.E...D. Bishop, _{London,, 1990.}

Ochi, M..K. and Motter, L.,

Prediction of Slamming Characteristics _{a.nd Hull}

Responses for Ship .Des'iqn', SNAME _{8,l 1973.}

27 -[ 9] Jensen,, J. and Pedersen, P.T.,

'Wave- induced Bending. Moments in

Ship- A Quadratic

Theory', Trans. RINA, 1978.

[1.0) Mansour, A. and d'Oliveira, J.,

'Hull Bending Moment Due to Ship _{Bottom Slamming in} Regular Waves', JSR Volume 19, 1975.

Kaplan, P.,

'Analysis and, Prediction of Flat Bottom Slamming impact of:AdVaflced;MarineVehjclesjnWaves, ?ISP Vol.34,1987.,

Keuning., J., ;..Distributj'on

of .. a Ship

University o1:: Technology, SHL., Report No. 817-P, .1988.

(13] Màtusiak., J. and Ratanen, A.,.

'Digital simulation of the Non-Linear _{Wave. Loads and}

Response of a Non-Ridgid Ship', CADMO 1986, Washington

D.C.

[14J Meyerhof f, W.,

'Added masses of Thin. Rectangular _{Plates. Calcu1atedfrom} Potential Theory', JSR Volume. 15,. .1970.

Petersen, J. and Marnts, L.,

'Comparison of Non-Linear Strip Theory Predictions and Model Experiments', PRADS 198.9,

yama.

Radev, D..,

'Numerical, Realization of problem _{for Entry of Rigid} Circular Cylinder in Compressible Fluid', IV th IMAEM Congress 1987, Varna.

Radev, D,

IV .th IMAEM. Congress :1987,Vamna.

.( 21 ].... Report.of Committee _iOth

.ISSC. 19:88., Denmark.

. Report of ..the. ITTC...Seakeepïng. :.Committee,

Proc., ofthe

19th IT.TC., 1990, Madrid.

Soares, C.,

'Transient Response of Shp Hulls to Wave impact', ISP Volume 36, l988.

Stavovy, A. .and.Chuang, S.,

'Analitical. Determination; _{of;.S1amrning ..Pressures-. for} High-Speed Vehicles in Waves',, :JSR 1976.

,Takemoto, H. _{, ..Hashiz.umi,,, ...Y. .and.. Oka., .S..,,}

...' Full-Scale.. Measurement.. of,. Wave_{:..Impact Loads. . and..Huli} ., Responces of a. Ship 'iflWaves',1.st...Repor,.jou.mna1 _of

the Society of Naval Architects _{of Japan, Volume 158,}

1985.

Wagner, V.,

'Überstosz- _{und gleitvorgänge an der Oberfläche von} Flüssigkeiten', Zeitschrift für Angewandte Mathemàtïk und Mechanik, Vol. 12, No.4, 1932.

-Appendix 1: input file of slamming simulation _program.

The name of the input file for slamming simulation program is INPUT.DAT. This file consits of the following input data.: MINDEG, MAXDEG, MZ3, MZI, NZ, NZ3, NZ1, _{Mxl, Mxl, NFl., N'F2,} NF3, NF,

BETA,, XG2O, YG2O, VER, Vii., ALPHA, DXY2, MTEMP, NPRI, KTIME., NSTEP,

Z (I), RHAS (i.), DA}!P (I), COR3 (I), 'CORB.(I).

Description of parameters.:

On. 'the.. first'row 'are::variabieg'::which;,_{show '..the approximation} degree..o.f

-.with M._{: are,, responsibleto.}

for f':damping, . starting

the coordinate system.. NFl '.isvresponsib'ie_{'for"the' frstthree'} terms in, formula (6') .NF2. jg for. the. fourth. term,. NP'3 is for the last .two terms., NF is for _{force F. The recommendation of} the author to use' four (4.) degrees _{for all variables.}

Approx-imation is done automatically _{for degree less than 4..}

Second row:

BETA _ß

.XG2O ' - XB coordinate of 'investigated 'point

YG2 O _B coordinate of .investIgated, point

VER

V

VH - U

ALPHA - a

DXY2, -

-expet coeficient. greater than 'limit where 'is

B _{applied strip»theory för wedge}

Third row:

MTEMP - number of approximation, points NPRI - number of printing points

KT1ME - number of points before' the _{gage touchs the water} NSTEP - Z.WSTEPZ * KTINE, where STEPZ is step in Z direction

The next rows: Z(i.), RHAS (I), _{DAMP(i), COR3 (.1), CORB(I)}

-Z(I) - array of vertical displacement RHAS(I) _{array of. sectional 2-D added mass} DANP(I) _{- array of sectional, 2-D damping}

BX

COR3(I) - J(, -.)

local reduction coefficient of added _mass

CORB(I) -

local reduction coefficient of damping

There are three cases of using _{the program.:}

- hydrodynamic coefficients are calculated by 2-D 'method (COR3(I) = i);

modified :on...thé.basjg 's

y O);

. .. -onthebagjs' of

linear' emp'iricaiapproximation('coR3

₍₁₎

_=..s:O'):..

-Appendix. 2: Output listing.

- 31

'P' ' 0.5720E-I-01 0.5291E+Oì. 0.4864E-i-01 0.'4463E4-O]. 0.41O.6Ef0'1

O.:3817E+o].

DEADRISE ANGLE: _{9.981830 DEG}

GAGE: X=0.010,Y=O.04O,Z= 0.007040

V vertical,: 0.72 N/S DT= U forward: 3.00 H/S DX/'DZ=

TRIM ANGLE: 2.50 DEG _DX/DY=

0. 0032907:0 -22.9038 4.28106 DX=0.. 053:69636 _DX/DT= DY=0.01332064 DY/DT= DZ=-.00234443 DZ/DT= 16.317621 4.047968 -0.712442 t 0.003863 0.007127 to -0.006538 -0.003274' 0. 0104.01 0.000000 0.013692 _{0.i 017004} 0.003291 0.. 006603 0. 02.0344 0.00.9943 Xo -0.019614 -0.009823 Yo 0.015799, 0.029120 Zo -0.002781 -0.005125 0.. 000000 0.. 042440 -0.. 007.469 0.009:8,72 _{0,. 019809} 0.055761 0. 069082 .-O. 009814 -0.. 012,158 0. 029829 0. 0824 02 -0. 0.14503 Xgagel -0.009.93.0 0. 000140 Ygagel 0.013359 Ò. 02:6679 Zgage1 o 004689 .0.002344 '0.009683 .0.0:19555 :0.029493 0.040000 '0..05332,]. '0.066641 :.00OO,OOO...'O.00'2344 ::'.O.004'689. 0.039512 0.07:99:62. -0.007033 Xgage2 0.010000 0.. 010000 Ygage2 .0.013333 0.026667 Z gage 2' _{0.002347 0.004693} .0..01o0o;o 0.04000.0 '0.007040 '.0 .01,00'O.o 0.'010000 '0053333 0.. 066:667 0.009387 0.011733 0.010000 0. 080 000 o. 014080 .XB2 0.063748 0'. 117495 YA 2 0.015814 0. 029.147 ZA2. 0.002783 0.. 005130 Lw 0.063808 0.117607 iO 171243 0.042.4:8]. 0.007477 0.171406 .0. 22499:0 0..,2 78 73 8 0:. 055814 0.069147 O. .00982 3 o o 012170 0.. 225204 0.. 279003 '0. 33:2:48'5 '0.082481 0.014517 0.332:802 Xgage3 0.053696 0.107393 Ygage3 0. 013321 0. 02664.1 Zgage3 -0. 002344 -0.00468.9 .0. i61089 0.039962 -0. 007033 0.214785 0.268482 0. 053283 0. 06660,3 -0.009378 -0. 011722 0. 322178 o.079.92.4 -0. 014067 DX/DY 3D 4.78 4.41 COR.3/2D 1.20 1.,10 VOL.3D 0.935 5. 86 'VOL.2D 1.67 13 3 Vundjnì. 0.560 0.439 4.28 1.0.7 18.1 45.1 '0.402 4.22 4 . 18 1.05 1. 05 41.1 78.. 2 107. 209. 0 3 85 O 375 4.16 1.04 '133. 3:60. 0.368 Speed X 2.965772 Speed z -0.849400 Vertical -o.. 719226 a _0.400411 2.965853 -0.847544 -.0.. 7173,69 0.738008 .2. 965983 -0 844577 -0. 714399 1.075607 2.966162 -0.84.0486 -0.710303 .1.413206 2.. 966390 2.966670 0.8,352:50 _-0.82.8844 '-0.705063 -0.698650 1.7508:03 2. 088402 m(s,t) .0.37.3:977 dn/ds -262. n (s , t') _5.494434 dn/ds -O.373E+04 . 1.2168.22 -455. 16.80062,9 -0.579E-i-04 2.498502 -637. 32.102139 -0.716E-i-04 4.193:650 -8.08. 4:9.993805 -0.803E+04 '6o2808'56 :8.742:67'S -971. -0.113E-i-04 69.480515 89.977203 -O.855E+04 -0.892E+04 11.4 dn/d'x 163. 19.9 . 2:53 27.8 313. 35:3 351. 42.5 4:9.3 373. 389. Pitia _-0.005621 PVvdin/d 7.09,22:69 Pnv 0.175178 Pl _0.5729E+O1 -0.. 016854 6. 13.4697 ' 0.26724.1 0.5375E-I-01 -OE 033:62:5 5.680990 0.339232 0.4'998E+0'1 -0. 055614 5. 356517 0.394304 0. 4603'E+Ol -0.082553 -0.114223 5. 086860 4.8474.35 0.435667 0. 466551 O ..4.197E+0]. O. 378 6E+0l Puvdm/dx -1.. 081939 P2 -.567.6E+0O -0.939226 -.6968E+0o -.0.8723:98 -.7544E+00 -0.826847 -.7513E+oo -0.790290 -0.758968 _.69:87Ef00 -.6076E+O.0 Psudn/dx -0.042469 -Psvdn/ds _0.278581 P3 _0.5611E+OEO -0.065969 0.431775 0. 613.9E+.00. -0.081645 0.532483 0.62.1 5E+ 0.0 .0 -0. 0914.90 0. 593765 .,,6104E+OO. -0 097497 -0. 101659: 0. 628760 0. 6505Ï1 0.. .6 o.7:ÏE+o O O .6381E+0Ø

DEADRISE ANGLE: 9.981830 DEG

GAGE: X=0.0l0,Y=0.040,Z= 0.007040

t

0.003863

to

-0.006538

Xo

-0.006538

Yo

0.015799

Zo

-0.002781

Xgagel

0.003145

Ygagel

0.013359

gage1

0.004689

Xgage2

0.010000

Ygage2 0.013333 Zgage2

0.002347

XB2

_0.063748

YA2

_0.015814

ZA2

_0.002783

Lw

_0.063808

Xgage3

0.053696

Ygage3

0.013321

Zgage3

-0.002344

DX/DY 3D

4.78

COR.3/2D

1.20

VOL.3D

0.935

VOL.2D

1.67

Vundjin.

0.560

Speed x

0.967676

Speed z

-0.762161

Vertical -0.719226

a 0.40041].

]u(s,t)

dm/ds

n(s,t)

dri/ds

dxn/dx

dn/dx

Pina

_-0.005621

Pvvthn/ds

5.710240

Pnv

0.157186

Pl

0.4644E+01

Psudn/dx -0.013857

Psvdn/ds

0.249969

P3 _0.5607E+00 DT=0. 00329070 DX/DZ=

_-22.9038

DX/DY=

4.28106

0.007127 -0. 003274 -0. 003274 0. 029120

-0.005125

0. 0064 09 0. 02 6679 0.002344

0.010000

0.026667 0. 004693 0.117495 0. 029147 0. 005130 0. 117607 0.107393 0.026641

-0.004689

4.41

1.10

5.86

13 3

0.439

0. 9 67757

-0.760305

-0. 7173 69 0. 73 8008

-0.016854

4.936790

0. 239733 0. 4352E+01

-0 021526

0. 387332 0. 6137E+00

0.010401

0. 000000 0. 000000 0. 042440

-0.007469

0. 009683 0. 040000

0.000000

0. 010000 0. 040000 0. 007040

0.171243

0.042481

0.007477

0. 171406 0. 161089 0. 039962

-0.007033

4 28

1.07

18 1

45.1

0.402

0.967887

-0. 757338

-0.714399

1. 075607

-0.033625

4.567992

0. 3 04 192 0.4034E+01 -0. 026643

0.477481

0. 6214E+00 DX=0.05369636 DX/DT= 16.317621 DY=0.01332064 DY/DT= 4.047968 DZ=-.00234443 DZ/DT= -0.712442 0. 013692

0.003291

0.003291

0. 055761

-0.009814

0. 012974

0.053321

-0. 002344 0. 010000 0. 053 333

0.009387

0. 224990

0.055814

0. 009823 0. 2 252 04 0. 2 14785 0. 053283

-0.009378

4.22

1.05

41.1

107.

0.385

0. 968065

-0.753247

-0.710303

1.413206

-0.055614

4.302259

0. 353 377 0. 3696E+Ol

-0.029859

0. 532 135 0. 6104E+00

0.017004

0.006603

0.006603

0. 069082

-0.012158

0. 016286 0. 066641

-0 004689

0. 010000 0. 066667 0. 011733 0. 278738 0. 069147 0. 012 170 0. 279003 0. 2 68482 0. 066603

-0.011722

4.18

1.05

78.2

209.

0.375

0,. 968294

-0.748011

-0.705063

1.750803

-0. 082553 4. 079745 0. 3 90163 0. 3342E+01

-0.031825

0. 563 088 0.6072E+00 0.020344 0. 009943 0.009943 0. 082402

-0.014503

0.019626 0.079962

-0.007033

0.010000 0.080000 0.014080 0. 3 32485 0. 082481 0.014517 0.332802 0. 322 178 0. 079924

-0.014067

4 16

1.04

133. 360,.

0.368

0.968574

-0 741605

-0.698650

2.088402

0.373977

1.216822

2.498502

4.193650

6.280856

8.742675

-262.

-455.

-637.

_-808.

_-971.

_-0.113E+04

5.494434

16.800629 32.102139

49.993805

69.480515

89.977203

-0.373E+04 -0.579E+04 -0.716E+04

_{-0.803E+04 -0.855E+04 -0.892E+04}

11.4

19.9

27.8

35.3

42.5

49.3

163. 253. 313. 351.

373.

389.

-0.114223

3.880716 0. 4 17444 O 2977E+01

Puvd/dx -0.316760

-0.274924

-0.255282

-0.241848

_-0.231024

_-0.221711

P2

_{-.l665E+00 -.2040E+00 -.2204E+Ø0 -.2l92E+00}

-.2034E+00 -.1764E+00

-0.033190

0. 582042 0. 6382E+00

P

_{0.5037E+0l 0.4761E+01 0.4435E+0l 0.4087E+01}

0.3746E+01 0.3439E+0].

V vertical: 0.72 H/S

U forward:

1.00 H/S

V vertical: 0.24 MIS _{DT=0. 00006667} U forward: 3.00 MIS DX/DZ= -114.5887

TRIM ANGLE: 0.50 DEG DX/DY= 0.51668

t 0.000430 0. 000497 0. 000564 to -0.000133 -0.000067 0.000000 Xo -0.000400 -0.000200 0. 000000 Yo 0.064540 0. 074540 0.084540 DX=0.00183335 DX/DT= 27.499777 DY=0.00999962 _{DY/DT=149.99j.730} DZ=-.0000].600 DZ/DT= -0.239987 0. 000630 0. 000697 _0.000764 0. 000067 0. 000133 0. 000200 o 000200 0. 000400 0. 000600 o 094539 0. 104539 0.114539 Zo -0.000103 -0.000119 -0 000135 -0.000151 -0. 000167 -0. 000183 Xgagel 0. 009599 Ygagel 0.010001 Zgagel 0. 000032 0. 009799 0. 02 0000 0.000016 0. 009999 0.030000 O 000000 0. 010199 0.040000 -0. 000016 o 010399 o 049999 -0.000032 0. 010599 0. 059999 -0.000048 Xgage2 0.010000 Ygage2 0.010000 Zgage2 o 000016 0. 010000 0. 02 0000 0.000032 0.010000 0.030000 0. 000048 0.010000 0. 04 0000 0.000064 0.010000 0.050000 0.000080 0. 010000 0. 060000 0. 000096 XB2 o 011833 YA2 0.064543 ZA2 o 000103 Lw 0.011834 0.013667 0. 074543 0.000119 0. 013667 0.015500 0. 084543 0. 000135 0. 015501 0.017334 0. 094543 0.000151 0.017334 0.019167 0. 104543 0.000167 0.019168 0.021001 0. 114543 0. 000183 0.021001 Xgage3 _0.001833 Ygage3 0.010000 Zgage3 -0.000016 0.003667 0. 019999 -0.000032 0.005500 0. 029999 -0. 000048 0. 007333 0. 039998 -0. 000064 0.009167 0.049998 -0.000080 0. 011000 0. 059998 -0.000096 DX/DY 3D 1.18 COR.3/2D 0.296 VOL.3D 0.263E-01 VOL.2D 0.640E-02 Vundini. 4.11 0.683 0.171 0.405E-01 0.512E-01 0.791 0.517 0.129 0.591E-01 0.173 o 342 0.433 o 108 0.826E-01 0.410 0.202 0.383 0.958E-01 0.112 0.800 0.140 0.350 0.875E-01 0. 147 1.38 0.106 Speed x 2.997792 Speed z -0.266167 Vertical -0.239997 a _0.014870 2.997792 -0.266166 -0. 239996 0.017174 2.997792 -0. 266165 -0.239995 0.019478 2.997792 -0.266164 -0.239993 0.021782 2.997792 -0.266162 -0.239992 0. 024086 2.997792 -0. 266160 -0. 239990 0.026390 m(s,t) 0.024706 dm/ds -0 258E+04 n(s,t) 3.815077 dn/ds -0. 34 6E+06 dxn/dx 22.7 dn/dx 0. 302E+04 0. 078715 -0 406E+04 10.853200 -0 528E+06 35.4 0. 461E+04 0.151610 -0 498E-I-04 20.542231 -0. 678E+06 43 . 4 0. 591E+04 0.236070 -0 554E+04 32 381695 -0.798E+06 48 3 0. 696E+04 0. 327859 -0. 593E+04 45. 939995 -0. 893E+06 51.7 0. 780E+04 0. 425828 -0. 634E+04 60.854404 -0. 968E+06 55 3 0. 845E+04 Pma _{-0 000018} Pvvdm/ds 9.156105 Pnv 0.050774 P]. _{0. 6487E-I-0].} -0.000034 7. 187898 0. 072222 0. 4230E+01 -0.000049 5.877584 0.091131 0.264 6E+01 -0 000064 4.902565 0.107739 0. 1737E-FOl -0. 000079 4. 198318 0. 122279 0. 1504E+01 -0.000094 3.743331 0. 134980 0.1949E-I-0]. Puvdm/dx -0.907016 P2 _-.6239E+00 -0.706381 -.4014E+00 -0.576811 -.2447E+00 -0.481952 -.1542E+00 -0.412828 -.1301E+00 -0.367386 -.1724E+00 Psudn/dx -0.007232 Psvdn/ds 0.073580 P3 0.1051E+0o -0. 011057 0. 112 496 0. 1650E+00 -0.014182 0.144292 0. 2073E+00 -0.016698 0.169886 0. 2354E+00 -0.018694 0.190194 0. 2525E+00 -0.020261 0. 206133 0. 2619E+00

p _{0.5973E+ol 0.3996E-f-0]. 0.2609E-Fol 0.1818E-Fol 0.1626E+0].}

0.2037E+0].

DEADRISE ANGLE: 0.091673 DEG

V vertical: 0.72 M/S _{DT=0. 00002222} U forward: 1.00 M/S _{DX/DZ= -114.5887}

TRIM ANGLE: 0.50 DEG DX/DY= _0.51668

DX=0.00183335 DX/DT= 82.517776 DY=0.00999962 DY/DT=450.075775 DZ=-.0000]600 DZ/DT= -0.720122 t 0.000143 to -0.000044 0.000166 -0.000022 0. 000188 0. 000000 0.000210 0. 000022 0.000232 O 000044 0. 000255 0.000067 Xo -0.000044 Yo 0.064540 Zo -0.000103 -0.000022 0. 074540 -0.000119 0. 000000 0. 084540 -0. 000135 0.000022 0. 094539 -0. 000151 0.000044 0.104539 -0. 000167 0.000067 0. 114539 -0. 000183 Xgagel 0.009955 Ygagel 0.010001 Zgagel 0.000032 0. 009977 0.020000 0. 000016 0.009999 O .030000 0. 000000 0.010021 0. 040000 -0.000016 0.010044 0. 049999 -0.000032 0. 010066 0.059999 -0. 000048 Xgage2 0.010000 Ygage2 0.010000 Zgage2 0.000016 0.010000 0. 02 0000 0. 000032 0. 010000 0. 030000 0. 000048 0. 010000 0.040000. 0.000064 0. 010000 0. 050000 0.000080 0. 010000 0. 060000 0.000096 XB2 0.011833 YA2 0.064543 ZA2 0.000103 Lw 0.011834 0.013667 0. 074543 0. 000119 0.013667 0. 015500 0. 084543 0. 000135 0.015501 0.017334 0. 094543 0. 000151 0.017334 0. 019167 0.104543 0.000167 0. 019168 0. 02 1001 0. 114543 0. 000183 0.021001 Xgage3 0.001833 Ygage3 0.010000 Zgage3 _-0.000016 0. 003667 0.019999 -0.000032 0. 005500 0. 029999 -0.000048 0.007333 0.039998 -0. 000064 0. 009 167 0.049998 -0. 000080 0. 011000 0. 059998 -0. 000096 DX/DY 3D 1.18 COR.3/2D 0.296 VOL.3D 0.263E-01 VOL.2D 0.640E-02 Vundjm. 4.1]. 0.683 0.171 0.405E-0]. 0.512E-01 0.791 0.517 0.129 0.591E-01 0.173 0.342 0.433 0.108 0.826E-01 0.410 0.202 0.383 0.958E-01 0. 112 o 800 o 140 0.350 0.875E-01 0.147 1.38 0.106 Speed z 0.993679 Speed z -0.728698 Vertical -0.719999 a 0.014870 0. 993 679 -0. 728698 -0.719999 0. 017174 0. 993 679 -0.728697 -0.719998 0. 019478 0. 993679 -0.728697 -0. 719998 0.021782 0.993679 -0.728696 -0. 719997 0. 024086 0. 993679 -0. 728696 -0.719997 0. 02 6390 in(s,t) 0.024706 dm/de -0. 258E+04 n(s,t) 3.815077 dn/ds -0. 346E+06 dm/dx 22.7 dn/dx 0. 302E+04 0. 078715 -0.406E+04 10.853200 -0. 528E+06 35.4 0. 46].E+04 0. 151610 -0.498E+04 20.542231 -0.678E+06 43 4 0. 591E+04 0.236070 -0.554E+04 32.381695 -0. 798E+06 48.3 0. 696E+04 0.327859 -0. 593E+04 45.939995 -0. 893E+06 51.7 0. 780E+04 0.425828 -0. 634E+04 60. 854404 -0. 968E+06 55 3 0. 845E+04 Pina _-0.000018 Pvvdm/ds 68. 627274 Pnv 0. 139007 Pl 0. 4830E+02 -0.000034 53 875450 0.197725 0. 3118E+02 -0. 000049 44. 054604 0. 249494 0. 1915E+02 -0. 000064 36. 746830 0. 294967 0.122 1E+02 -0. 000079 31. 4 68508 0. 334776 0. 1037E+02 -0.000094 28.058464 0. 3 69550 0. 1362E+02 Puvdm/dx -0.823099 P2 -. 5662E+00 -0.641030 -. 3642E+00 -0.523449 -.2221E+00 -0.437367 -. ].400E+00 -0.374640 -.].180E+0O -0.333404 -.1565E+00 Psudn/dx -0. 002397 Psvdri/ds 0.201443 P3 _0.3154E-i-00 -0.003665 0. 307986 0. 4951E+00 -0.004701 0. 395039 0.6220E-i-00 -0.005535 0.465111 0. 7062E-i-00 -0. 006197 0. 52 0712 0. 7575E+00 -0. 006716 0.564352 0. 7857E+0o

P _{0.4805E+02 0.3132E+02 O.1955E+02 0.}

1278E+02 0.11O1E+02 0.1425E-4-02

DEADRISE ANGLE: 0.091673 DEG GAGE: X=0.0].0,Y=0.030,Z= 0.000048