Non linear heave and pitch motions of fast ships in irregular head seas

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Deift University of Technology Ship-Hydromechanics-Làboratory Mekelweg 2 2628 CDDelft The Netherlands Phone 015 - 7868 82

NON LINEAR HEAVE AND PITCH

MOTIONS OF

FAST

SHIPS IN

IRREGULAR HEAD SEAS

Ir. J.A. Keuning

Report No. 918-P - June 92

Intersociety High Performance

Marine Vehicle Conference and

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24 through 27 JUNE 1992

R1TZ-CARLTONHOTEL

ARLINGTON, VA

INTERSOCIETY

HtGH PERFORMANCE

MARINE VEHICLE

CONFERENCE AND

EXFÍIBIT

Proceedings:

(4)

(5)

PROCEEDINGS 0F THE

IN1rERSOCIETY

HIGH PERFORMANCE MARINE VEHICLE

CONFERENCE AND EXHIBIT

HPM\T '92

24 through 27 JUNE 1992

RITZ-CARLTON HOTEL

ARLINGTON, VA

A

instantaneous submerged area abf Büo'ancy correction factor

a113 Significant acceleration

'acG1/3 Significant acceleration at CG

abwll3 Significant acceleration at the bow Project area

B Breadth of ship

b Instantaneous half-beam of ship b Half beam from centre line to dhine CD, c Cross flow drag coéffiòient

NON LINEAR HEAVE AND pFCH I»K)TIONS OF FAST SHIPS IN IRREGtJLAR HEAD SEAS.

by ir. J.A. Keuning Deif t University of Technology Ship Hydromechanics Laboratory Report No. 918-P March 1992

D d D/DT dF Fx Fz F0 FL g h I 'a k ka L 1 l M Ma ma NL r r0 T t u 'V W WZ X X XCG Xd Xp 5CG Za Zal/3

Drag, resistance of ship Depth section

Time rate of change Force per strip

Dynamic force per strip Buoyancy force, per strip

Hydrodynamic force in x'- direction Hydrodynamic force in z-direction Hydrodynamic moment about CG Lifting force in e-direction Gravity acceleration, 9.81 ms2 Height of water from base line to waterline n z-direction

Ship's moment of inertia

Instantaneous moment of inertia of. added mass of the ship

Wave núniber of wave component Added mass correction factor Ship's length

Ship's length

Wetted length at the chines Wetted length at the 'keel Mass of ship

Instantaneous added mass of ship Instantaneous added mass per strip Lifting moment about CG

Static moment of added mass of ship Instantaneous wave height per strip Amplitude of wave component

Trus,t or towing fOrce of vessel Instantaneous time

Velocity parallel 'to base line Velocity perpendicular to base line Weight of ship (N')

Orbital water velocity

X-coordinate State vector x-coordinate of CG Leverage of drag to CG Leverage of thrust to CG z-coordinate of CG Heave amplitude

(8)

strip theory" approach. This theory has been adequatly described by Tasai (1S, Ursell

[19') and Gerritsma and Beukeiman (8). It is a linear theory which implies that the ship moving in waves is considered to be a linear system as well as infinite small motions a-round a still water reference position which are linear related to waveheight of the in-comning wave.. The hydrodynamic coefficients and wave forces of the ship in its still wa-ter reference position are being calculated by integration over the length of the ship of 2-D cross sectional values for added mass and damping, derived either by some kind of conformal mapping or a 2-D diffraction tech-nique. The speed influence is taken into ac-count by a correction incorporating a.o. the

lengthwise distribution of the added mass

and damping over the length of the model.

Typical restraints of this calculation

method are:' its linearity, moderate forward speeds, moderate motions, slender hull forms and wave lengths in the order of 0.5 to 3.0

times the sh.iplength. The use of this theory' is widespread, its applicability is high and the results generally good. The restraints mentioned however should make' the applica-tion of this' theory not, justifiable for fast ships.

Recently Blok and Beuke'lman (1] however showed "that the linear strip theory yields

reasonable results for a round bilge hull

form for speeds upto Fn = 1.1 as far as the

heave and pitch motions are concerned. A typical example of the result they derived is shown in Figure 1 and Figure 2..

I" 'o o 1.5 '-o 05

Figure 1. Pitch transfèr function head seas.. From (1].

This lead Beukelman (3] to use this method for the calculation of the heave and pitch

notion and vertical accelerations of a

planing hard chine hull form in waves and to

compare this with model experiments. in

MODEL 5 Fn. 1.140 EXRFIT

___

_...-.._CLOSE o MARIN

\

ß Deadrise angle

pl Deadrise angle of upper hull Angle between trust and drag

r Coordinate in ç-direction

Wave amplitude

.ai/ 3 Significant wave amplitude

e instantaneous angle between undis-turbed water level and base. line, of vessel

Pitch amplitude

ea]3

Significant pitch amplitude

A _LIE-ratio

V Wave slope

r 3.14159

p Specific mass of water 1.025

q' Random phase of wave component 3 Dimensional correction factor

w Radial frequency of wave

V Displacement of ship

INTRODUCTION

In the last decades the interest in fast

marine vehicles shows a continous growth. Typical applications for these kind of high speed vessels are 'among. others: patrol boats, combattant ships, passenger ferries and pleasure boats. Traditionally the opera-bility of these vessels was constrained to the more or less sheltered areas of rivers,

lakes and coastel seas in which the wave

climate generally may be described as mild. The combination of high forward' speeds and acceptable motions in waves using a mono-hull, with regard to passengers comfort and safety as well as structural loads, proved to be difficult and in search for this com-bination all kinds of so-called "advanced" concepts have been designed, evaluated, build and used, every one. of them with its particular benefits and shortcomings. Among these the Hydrofoil Ship, the Air Cushion Vehicle, the Surface Effect Ship., the Small Waterplane Area Twin Hull Ship, the Catama-ran and the Wavepiercer should be mentioned, all of which have reached a certain degree of perfection. Compared to the. Monohull, be it a round bilge or a hard chine design, sIl these "advanced" concepts tend to be more complicated and more expensive and therefore

the role of the relatively "simple" fast monohull is not finished (yet). The aplica-bility in waves of this concept however needed improvement.

Improving the operability of the fast mono-hull meaned improving the seakeeping

behav-iour of the concept in particular in head

seas.

An important aspect in the optimisation i the availability of an adequate motion cal-culation routine capable of predicting the motions of a fast monohull in a seaway. The generally used calculation methods for

the motions of monohulls with moderate f or-ward speed in waves, both regular and Irre-gular', are based on the so called' "linear

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1.5 1.0 0.5 o 1. I 1.0 0.9 Za/ b.8 0'.? '0.6 o:s 0.4 0.3 0.2 '0.1 0.5

'fEì

Fu, 0.124 e [ca la Led 0 measured 10 RODEI. NO. 3 15

'generai 'he found good agreement for heave

for speeds upto Fn O .9 in both regular and' irregúlar waves, althoùgh no information is

available on peak valües of the motions

and/or accelerations. The agreement for

pitch, and in particular for the vertical

acceleratibns at the bow were less satisfac-tory. Typical results of his work are shown in Figure 3 and Figure '4.

Figure 3'. Heave transfer function planing 'hull. From [3]' 1.2 1.0 0.9 'Ca 0.7 0.6 0.5 0.4 o.3 0.2 o.1

Figure 4. Pitch transferfunction planing hull. FrOm' (31,.

For the purpose of a quick assessment of the operability of a fast monohuil in waves the

linear strip theory is a quite attractable tool 'due to the relative short processing

time needed for the computations and the

fact that motions in various spectra may be easily calculated using the linear superpo-sition principle once the transfere func-tions are known. That may well explain its widespread use in;optimisation studies.

From bot'h the real life experience and' model experiments however it is well known that the motions and in particular the vertical accelerations of a fast momnohü:Il in' head seas may be strongly non linear'. his non linear behaviour is already demonstrated by many authors. For instance Van' den Bosch (4] who found' strong non linear distributions of maxima in' the, bow vertical accelerations dependèd on the 'bottom deadrise angle. See Figure 5 and Figure, 6.

percent 20

io

0 0.2 Vn 0.724 calculated O measured MODEL NO. 3 s -L 0.4 0.6 0.8 1.0 1.2

l4

T7ç

100 80

I

20 2 t

6 8 10 12 14 16 1820 >20

af rnsec2

Figure 5. 'Frequency distribution of bow ver-tical, accelerations for ß 12.50. From ['4] MODEL 5 Fn. 1.140 1 2 FIT EXP VERSION VERSION CLOSE o MARIN

r'

o!'

o

Figure 2.. Heave trànsfer From [1)

f unc t ion head seas;.

(10)

MODOS = 25.00

loo

80

60

Fn

27

40 L/2çai13;Z1Li./;

i

20 o--

1__.

0 2

4

6 8 101214161820 >20

.af rn/sec2

Fgure 6. Frequency distribution of bow ver-tical accelerations for ß = 25° From [4).

The. simplification of the system of a fast moving monohull in waves to a linear system

demands too much "trimming off" of important physical phenomena.

Sources of the non linear character of the system may be f ond a.o. in the change of reference position of the craft (sinkage and

trim) düe to its high forward speed, the

pressure distribution over the bottom of the vessel at speed, non linear added mass du

to the changing geometry of the craft in contact, with water while performing large motions, non linear lift and damping and non linear wave exciting, forces also caused by the relatively large motions.

This lead to the development of non linear 'mathematical models for the calculation of

ship motions at high forward 'speeds.. One of the well known models was 'described by Zarnick [20) for heave and pitch in head regular waves, this model has been used as a

basis for the deve1opment of an inproved

model used in the present study. It was been extended to perform motion calculation in

irregular waves and various modifications

have been implemented with respect to the. calculation of trim and sinkage., added mass., wave- and bouyancy forces. All these aspects will be dealt with in short in the following paragraphs.

Due to the non linear character of the ma-thematical model the equations must be. solv-ed in the time 'domain which implies rather long processing times and' new simulations of adequate duration for each new spectrum. For the. optimisation of à design with respect to operability which implies the use of a scat-ter diagram and thereföre calculation of the motions in various spectra the procedure is quite cümbersome.

The scope of this study however is to show that this may be a inevitabel effort because

linear calculations may yield erronous

results and even' reversed trends.

NON LINEAR MATHEMATICAL MODEL FOR HEAVE AND PITCH OF FAST SHIPS.

The non linear, model used' for this study is based on the work of Zarnick [20). He f ormu-lated 'a non linear model for t'he heav,e and

pitch motion of a hard chine planing hull

with constant deadrise over .the length of the. ship in regular head' waves. The mathema-tical model is a non linear strip theory

approach, deviding the hull over the length in number of transverse strips with constant

cross sectional shape. A short summary of

the mathematical model will be presented

here.. 'For a more 'detailed description of the model reference is made to [13).

The coordinate system as presented in Figure 7 will be used, representing the vessel in its steady state equilibrium position at speed.

Figure 7. Coordinate system.

The equations of motion for heave,, surge and pitch according to Savitaky (16) become':

surge: MCG = Tcos(O+E) - NsinO - DcosO heave,: MZCG Tsin(O+E) .Ncosû - DcosO

pitch: IO Tx + Nxc - Dxd

in which:

M = the,ship mass T = towing force

N total hydrodynamic and lifting f orces

D - Drag

by assuming a constant forward speed theac-celeration along the X axis may be con-sidered equal to zero. For the present 'study this simplification will be used. Although omitance of this restriction is a foreseen future development.

The total lifting force on 'a strip 'is de-scribed by:

D

dF = (It1aV) + CD,cp'bVZ - afpgAcosO

in which:

dF = the force per strip

ma = the added mass of the strip V = the vertical velocity in the

plane of the cross section CD,_C = the cross flow drag coef f

i-dent

b instantaneous half beam of

the strip including pile up' of the water

p ercent

F°

20

(11)

r ama I

Uy d

i

a dV + Urna + dC

-

coso sino JmawzdC

_kapi

PS1CD,

cbi2dC

Uy-dC db

jVb - dC

+ 1

dt

pg

r I

aFAd

cosOl

-

cosûsinOf mawzCdC.

-

kaplrj Vb

CdC +

j

r

CdC - pS1CD,cbv2 CdC

-pg r

+ cosoh1lC

Ma

J1ma

mCdC ADDED MASS

As can be seen from this formulation the

added mass and its distribution over the

length of the. ship play an important role in

the determination of the dynamic lift.

In the present mathematical model the added

mass calculation is based on the

instantane-ous submerged beam. of the

cross. séction,

correctéd for pile up of the water., using

the Wagner formulation for the frequency

in-dependend added mass:

ma, p7rbZka

in which ka jS a constant for each section,

dependend on deadrise

and beam

to

draft

ratio of the particular section under

con-sideration. By doing so the. restriction

i

constant deadrise of the hüll has been

ehm-mated. The determination of the value of ka

may be based on the work of Uwang (6.]:.

By using the instantaneous beast of thé

sub-merged section for the.determination of the

added mass, the magnitude. hereof-becomes

de-pendend on the amplitude of

the combined

heavé and pitch motion as well as dependend

on

the

change

in geometry

of

the

cross

section. this is a considerable non linear

effect in the mathematical model.

The added mass formulation enabling to do

this is the frequency independend Wagner

ap-proximation. This approximation is generally

considered to be valid for high frequencies

onhy. This assuntion is however jüstifiabhe.

in the situation under consideration of

a

ship moving at high forward speed against

the waves:.

Keuning (1h] however showed that in the case

of a round bilge hull at relatively high

f orward speeds the added mass is practically

frequency

and

speed

independend

even

at

relatively.

low frequencies

if- the actual

pressure distribution over the length of the

ship due to high forward speed is taken into

account.

He performed forced oscillation tests with a

segmented model of a semi displacement f

rig.-ate type hull form, the sanie model as used

by Blok and Beukeiman.

These oscillation

tests have, been carried out with the model

in the proper reference posItion with regard

to sinkage and trim corresponding to

the

forward speed of the model1

i.e. .Fn

0.-57

and Fn = 1.14. This proved, Lo be an

impor-tant parameter.

The

pressure -distrIbution

over the length of

- the. model; due. to

the.

forward speed -and the change in this

pres-sure distribution due to the change in

sub-mergence of the section caused by the

oscil-iatory motion have been measured in a quasy

steady way.

The results. of these

méasuré-ments have been used to determine the added

mass of the sections by elaborating the time

histories of the force- signals obtained in

the oscillation tests. A graphical

prensen-taLion of one of the resùlts of this may be

seen

in

Figure

8,

in

which

figure

the

results

of

the calculated results

of

the

added mass are presented also.

The value of the cross flow drag coefficient

CDC used in the expression of the dynamic

abf

bouyancy

correction

coef f

i-cient

A =

instantaneous submerged

scv-tional area

g =

acceleration due to gravity

U and V are the velocity components along

the length of the hull, resulting from the

coìnbination of the forward velocity of the

vessel, its heave and pitch motions and the

wave

orbital velocities,

in

the

arjrl O

direct ion respectively.

These can be expressed by:

U _iCCGCOSO _-

_{CG - wz)SinO}

= + CG - Wz)CoSO - OC

The dynamic lift is considered to originate

from the change of momentum of the oncoming

flow and a cross flow drag force on the

sec-tions due to the vertical component of the

velocity.

Thè

elaboration

of

the

change

of

fluid

momentum yields:

r

dv

din dV din

I (-me. -. V

a

+ Urna - + U'J

a

dt

dC +CD cPb\T2 ) SiflO dC dV

_a

dV =

-dt

dt

dC d

+CD,cpbV2)cosO

- afpgA)dC

dV din F

U -rn --V--d

1

dt

+CD,cbV2)

- afpgAcos0)Cde

Zarnick

(201

elaborated the following

equa-tion for the lift on the hull:

F,L -O(iCCGCOSO

CG0)Ma + cosO f mawzdC

(12)

lift is determined using the work of Shuford (181 for V shaped sections with deadri;e

angle

fi:

CDc 1.30 cosß

A more detailed description of the modifica-tions to the mathematical model used may be found in Keuning [13].

REFERENCE POSITION OF THE VESSEL AT SPEED An other important aspect in the proper cal-culation of the. motions of planing craft is introducing intó the. calculations the actual position of the craft due to its forward

speed as a' reference position. Due to its high forward speed the craft experiences a certain trim and sinkage with respect to its original position at zero speed. This sink-age and trim causes a considerable change in

the geometry of the submerged part of the

Craft with respect to that zero speed. This

is an important deviation from the linear

strip theory mathematical models with zero speed' position is used in the calcúlations for the added mass, damping and wave excit-ing forces.

In the present mathematical model the actual position of the craft at speed is

calcu-lated using polynomial expressions for

resistance, sinkage. and trim derived from a extensive study based on a large number of experiments with models of planing craft,. This work was originally initiated by Clement and Bi' aunt (5.]..with..their well known systematic series of planing , hull forms derived from one parent hull form with 12.5 degrees of deadrise. They varied the length to beam ratio of the model resulting 'in 5 models with a range of length to beam ratio from 2 to 7. All these models have been

tested with 4 different weights and 4 dif-ferent longitudinal positions of the centre of gravity. The parameters that have been varied in the series were:

Length to beam ratio L/B

2-7

Longitudinal position of

the centre of gravity LCG ' 0-12 of li

Loading factor A/V2/'3 4 - 'a

In the search for an improved seakeeping

behaviour of the planing craft it was demon-strated 'by many authors that 'an increase in deadrise resulted in a benificial effect on the motions and the vertical accelerations in particular.

Therefore Gerritsma and Keuning ['10')

extended the original series of Clement and

Blount with two new series with 25 and 30

degrees deadrise angle respectively.

The parent models of these.new series were

derived from the original parent model as

used by Clement and .'Blount by keeping the vertical projection of'. the':pianform of the'

chine and deck identical and using the same length to beam ratios, loading factors and

longitudinal positions of the centEe of

gravity. A bodyplan of the three parent

models is presented in Figure 9.

By doing so an extensive systhematic series of 240 different models was created from 'all of which the resistance., sinkage and trim

has been measured in a speed range from

volumetric Froude numbers ranging between 0.5 and 3.0.

Although primarily intended to be used as a design tool for assessing the trade off

between improved seakeeping behaviour and

resistance in the design of planing ships, the data obtained for sinkage and trim proved to be an important result as well. correspond closely to the values presented

by Shuford (18] for planing ships in the speed range under consideration.

Table 1 length m 26.25 15.00 beam m 6 .12 4 . 41 deadrise degr. 25.00 12. 00 displacement m3 83.60 27.30 Fflv 1.90 2.70 ka 1.25 1.12 abf O .74 0.70

moasu red

000 cjilcul.tcd

Figure 8. Added mass along the length of segmented body at high forward speed. From (11].

BOUYANCY FORCE

Due to the dynamic lift, and the. flow

separa'-tion over part of the chines and at the

transom the buoyancy force needs a correc-tion on the straight forward hydrostatic

buoyancy force calculation using the

sub-merged area of the cross sections. in the present study this correction coefficient abf is assumed to be constant over the length of the ship. The value of abf is

determined using the known trim and sinkage of the ship at speed in combination with the given longitudinal position of the centre of gravity of the ship.

The equations describing the equilthriúm condition of the ship in this steady motion may be derived from the presented equations of motion with z and O fluxes equal to zero.

Using estimated values for CDC and ka the value of af may be determined by solving

these equations.

A typical result of this calculation proce-dure is presented in Table 1 for two dif-ferent boats. The calculated values of a

(13)

Results of the application to an other

design not belonging to the systhematic

series used for the derivation of the poly-nomial expressions, _{i.e. a patrolboat, may} be found in Figure Ii.

This result is typical for other verifica-tion calculaverifica-tions carried out as weil.

R410T46 r,;,, O Tfl.Io.eap. 0.0 0.3 1.2 1.6 ío.(Ocp) (J * Theta A. t That, A, tøt R410T46 RiSC CC

Figure 11. Comparison of trim and .sinkage of model not belonging to systematic series.

The improvement over the original formula-:tion of Savitsky used by Zarnick is of sig:-nificance and appeared: to be of importance.

This may be demonstrated by the results given in Tàbie 2, in which the inflüence of an error in the sinkage and trim calculation

on the resulting motions is presented. Table 2

8

WAVE EXCITING FORCES

One other source of strong non linear behav-jour of the system may be found in the wave exciting forces. In assessing the limits of operability of a high speed vessel one is more interested in extreme motions and peak accelerations than in the linear extrapola-tion of, behaviour derived from calculaextrapola-tions with small motion amplitudes.

This implies that the. effect of large

rela-tive motion amplitudes of the craft with

respect to the wave elevated water surface have to be taken into account in the mathe-matical model.

In the dynamic lift calculation Zarnick al-ready accounted herefore by using the in-stantaneous beam of the submerged sections...

Above the chine however he too assumed a

vertical prismatic extension of the free-board. This restriction has been omited in the present calculation roütine by defining. the actual hull geometry above the chine as well.

Keuning (131 performed wave force

measüre-ments with a restrained model at high

forward speeds in head waves. IÌhis meas-urement he used the same segmented model as

previously used in the forced òscillation

tests. The same speeds have been used as in the oscillation experiments, i.e. Fn = 0.75 and En = .1.14. The model was fixed. in its.

proper reference position with respect to

sinkage and trim corresponding to the for-ward speed under, consideration The

intro-duct-Ion of the proper reference position

proved to be important for deriving accurate results also. Two differeñt wave heights

have been used during the experiments and

the wave lengths varied corresponding to waveiength/sh.iplengt'h ratios from 0.6 to 3.0. The results of these measurements nd acompaning calculations revealed that in the region of interest the wave forces were dominated by the Froude Krilof f cOmponent.

This is an important conclusion if non

linear wave forces are to be calculated in a time domain solution, in which no frequency effects can be taken into account in an ir-regular sea. In particular if the pressure integration of the undisturbed wave was performed over thé actual "wetted't surface of the hull. Diffraction effects were of minor importance when compared to these non linear, aspects brought into the Froude Krilof f wave force. calculation. It should be noted of f course. that.the. change in geometry of the particular hull under consideration n the area hItouched by the .waveeievation determines to a large extend '.the.magnitude of this non linearity. A typical result of these measurements is presented in Figure 12.

in the present mathematical model this Froude Krilof f wave force calculation, in which the pressure of the undisturbed wave is being integrated over the actual area of the moving, heaving and, pitching ship in contact with the: water, has been extended with respect to the original formulations used by Zarnick. The geometry of the hull above the chine and in particular of the bow

sections proved: to be of importance when

predicting peak vertical accelerations at the bow.

H

7 H

'

.7

.j___.__1____._j_____J__._J L I l____t__._J __J_____l__

A _a1/3. zal/3 Lj*Oal/3«avl/3 acqlj3

al/3

.,

al/3 'al/3 a1/3

Fflv = 2.7 5.6.6 1.22 113 176 95 AO = 10 5.66. 1.41 127 191 114 Fnv 2.7 5.66. 1.22 113 176 95 Az 0.01. 5.66 1.27 115 153 92 0.8 1:2 I;G 2e 2.3 r (-.1

O ttCGa.p. * RCA A O RCA h. loi

0.4

0.3 0.3

(14)

Figure 9. Body plans

of

parent models with

12.50, 25° and

30° deadrise.

Ali the data of this systhematic series have been elaborated to yield poÏynómial expres-sions for resistance, trim and sinkage. The varIables used in the polynomial expression are:

ß

Lia

Ar/V 2/3

LCG

yielded far better results than the more

usual method of trying to find one polyno-mial expression covering the. wholé speed range.

The coefficients AO to AlO have been deter-mined using a least square curv fitting, method. The coefficients of the polynomial expression may be .found in Keuning' (10). A typical result of the fit of the' ,po1'nomial expressiOn to the' original data may be found

in, Fïgure. 10.

[cxc hip

libo CG

-7

---ois t

deadrse at midship section

I 6 18 2 22 24 26 2

length to beam ratio loading factor

longitudinal position of the centre of, gravity

351 2.5 2' rn(d,pl) (-J U 0Cc ,p . t RCG 0 loi 8 RCGAp Lexc hip

Figure. 10. Comparison of trim and sinkage measurement and calculations for model of systematic series.

9° 25,Fn' 2.5

67 08 09 010 411 ' o12

o 4a964E-02 -1.5246E-02 56253E-04 &.9504E03 hi3180-0I S.6000E0I

fcc 1.3219E*01 l9980E01i l3029E026l592EOl6!4i60*0Q 9i844E-OI

¡3 25 Fn =2 5 D' aZ a3 g o, Zj 3.4573E 01 9.0955E*02 -2.4403E'OO -.1.93100+02 3.5959E-02 Z .31390 401 3.1356E-03 -1.21110+00 -ß.4050E*00 -I.I03E002 9.7490E-01 i 877oE+oi -42799E-02 -8.5573E-01

Only the polynomial expressions for trim and

sinkage which have been derived from the

data will be. déalt with here. They have the following shape:

O (Fn,)

= a0+a1 (L/B) +a2 (L/B) 2+a3 CL/B) ZCG(FflV)

+ a4Ap/V2/3 + a5(Ap/72/3)2 + a6p/2/3.)3 + a7(LCG) + a8(LCG)2 + a9(LCG)3

+ a10(Ap/V2/3)(LCG) a11 (A/v2"3) (LIB) + a12(LIB).(LCG)

They are derived for 10 different volumetric Froüde numbers the range from Fnv = 0.75 to

Ffl - 3.0, i.e. for Fnv 0.75, 1.00, 1.25,

1.50, .1.75,, 2.00, 2.25., 2.50, 2.75 and 300. This technique of using different po) ynomial expressions for différent Froude numbers

rn(dpI) J-.)

O I.in,.o.p * TomA=,ioi 8 Inn, 6

0.5

o I I I 'I I I

(15)

1090

measurement

'°°v>

calculation

Froude-Kriioff force

Figure 12. wave forces along the length of ¿t fast moving model, Fn = 1.14.

From ('13.] IRREGULAR WAVES

The extension of the program to irregular

waves has been achieved by implementing a

subroutine which calculates a time history of the wave elevation and corresponding orbital velocities in an irregular sea and is described by Kant [9). The wave spectrum in which the motions of the vessel have to be calculated is discretised in a given

number of f requency intervals with one

centre frequency and a wave amplitude which results in the same energy as for the

inter-val of the specific spectrum, i.e.: N

ç(x,t) 00 _{Ça(i)Smn(wit + kjx +} _j)

The spectrum shape nay be defined both by

specific information of one particular spec-trum or by an energy distribution over the frequency range as formulated by

Brettsnei-der or Pierson Moskowitz requesting as in input a significant wave height and zero

crossing period. This latter approximation

is particular usefuil for optimisation of

the vessel when a large number of different wave spectra have to be simulated for asses-sing the operability of the vessel.

Using the well known dispersion relation for deep water waves a time history is generated' using a random phase generator for the dif-f erent components.

in the calculation of the forces in the present mathematical model for the calcula'-tion of heave and pitch mocalcula'-tions not only the wave elevation in the centre of gravity must

be known but also the profile of the wave

and the orbital velocities over the entire length of the vessel, as may be seeñ from

the formulas used. By utilising the tech-nique described the wave is known both as a function of time (t) 'as well as of a func-tion of place (X). In the simulafunc-tion program the vessel is actualy moving with a given' speed, against these waves, yielding the wave

profile and orbital velocities and their

distribution over the length of the vessel at every tIme step used in the calculations. By proper selection of the frequency bands used' the repetition time of the generated signal can be chosen considerably larger than the simulation time used in the calcu-lations. This method closely resembles the' situation in a towing tank when generating irregular waves.

VALIDATION OF THE MATHEMATICAL MODEL

The complete mathematical model has been

elaborated into a computer source code writ-ten in FORTRAN 77. A considerable part of this work has 'been done by a number of gra-duates from the Shipbuilding Department of the TU Deif t as part of their student thesis work. In this respect the contributions of

).C.2... _Verkerk _(1987), _Kring _(1990), _Kant ₍₁₉₉₀₎

and Quadvlieg (1992,) should be mentioned.

9

In order to be able to validate the outcome of the calculations using the mentioned com-puter source 'code, Keuning ('13] performed an extensive series of motion measurements with three planing craft in irregular waves. The work was commissioned by OElO the Netherlands.

For the experiments he used the three parent

models of the

Clement-Blount-Gerritsma-Keuning systematic deadrise series. The mo-dels had a deadrise angle of 12.5, 25..0 and 30.0 degrees respectively. The experiments have been carried out in the Deif t Shiphy-dromechanics' Laboratory. In the experiments three different wave spectra have been used, defined significant wave height and peak period according to the data in Table 3: The models were tested at two different speeds, corresponding to volumetric Froude numbers of Fnv = 1.65 and Fn 2.70.

The influence of the deadrise on the motions

QQ

6686ó6o5o

R

QQ

6 6

o 666666e e

s O

i666

6 60 o a Q a

00)0000

0 e We s

____

'Ç

,0' o 4... 2o.e o 4... 2... o 4000 2,00 o 4900 1000 o 4IO is. 1' 5,0 ,.. is. S.C.,

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Ship Length 15 meters.

and vertical acceleratiàns is clearly demon-strated in Fïgure 13 and the relation be-tween the significant values and the

measur-ed peak valües in Table 4. The non linear

behaviour in particular in the vertical accelerations at the bow is once again clearly demonstrated by the outcome of these experiments. At the same time however this nonlinear behaviour appears to be dependend

on the deadrise angle of the vessel undér

consideration. The results of the calcula-tions with the present mathematical model are presented as well. Geñerally spoken the correlation between the measurements and the

calculations is satisfactory although

discrèpancies do occur 6 u G) w Q

significani

G)LCd

Table 3

Figure 13. Measured and calculated signif i-cant and maximum bow vertical accelerations as function of deadrise.

Table 4

OPERABILITY, ANALYSIS USING LINEAR VERSUS NON LINEAR MODELS

As indicated before the availability of an

adequate motion calculation routine is

available tool for performing optimisation

work on fast ships with respect to their

operability in waves.

To be able to assess the operability of fase vessels in waves the first need is for a set of appropriate criteria with respect to the limit of motions and vertical accelerations acceptable for a save and/or comfortable operation of the ship:. In order to obtain these the Shiphydromechanics Laboratory of the Delf t University of Technology carried out a number of real scale experiments aboard of fast ships at sea under command of the regular crew. The ships. on which the

ex-periments were carried out ranged in size

from 16 to 35 meter length over all and speeds yarned in the range from 20 to 30

knots.. Although all the ships had a specific task where they were designed for i.e. patrol boats, pilot launch e.a., which

gene-rally imposes limits on the amounts of

motion and/or accelerations acceptable in order to b able to perform their specific task designed for, in this study those were

not considered. General information was

sought on what limits., the save, and comfor table passage of the ship trough the waves. The crew were asked to maintain a speed as high as tought to be acceptable against the waves. Both the motions and accelerations at differeñt places along the length of the ship were measured, as well as the waves. Important conclusions to be drawn from these experiments was that generally spoken the

crews tended to imply a voluntary speed

reduction at roughly the same conditions and that not so much the significant value of both motions and vertical accelerations were a measUre of this voluntary speed reduction but much more the occurence of one big slam with associating peak in the vertical

acce-lerations at the bow.

Spectrum H'ai/3 avi/3

J

r

I 2.52 2.37 5.7 I Ffl 1.62 II 5.34 3.91 8.3 I 118.05 4.20 15.9 12.5° 1 2.27 6.04. 18.0 I Fn 2.70 II 5.45 10.86,48.0 L 111H7.90 11.44 55.0 I 2.07 2.19 5.0

Fflv =

1.62 II 5.16 4.09 9.5 111H7.25 4.23. 11.8 25° , T' I 2.28 4.20,10.l Fn = 2.70 II 5.02 7,.3l,22.5 L III 7.98 ' 8,.2630.0 I 2.28 2.41 4.2 Fn = 1.62 II 5.58 4.43 8.6 III 7.96 5..02L10.2 30° J.

Ï

2.41 4,47 ..4

L Fnv =

2.70 II III. 5.76 7.95 8..:04' 9.4923.l 19.6 Spectrum I H113 = 0.55 rn T0 = 5.9 sec Spectrum 2

H113 =

1,10 n T 6.7 sec

Spectrum 3 H11i3 = 1.60 m T = 9.0 sec

0 5 tO IS. 20 25 30 35 Dedrise D IO IS 20 25 30 35 De ath is e 10 0

u6

G) w

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As soon as this happened the crews reduced speed in order to avoid it' happening again irrespective of prevailing significant (average) motions at that time.

in order to be able to set criteria which were usable at that time for use in

optimi-sation routines based on linear tools for motion calculations, the Dutch authorities however formulated criteria based on

igjii-f icant values igjii-for vertical accelerations at midship and at the bow.

Table 5

The basic underlying assumption behind this is that 1f the waves are supposed to be

Rayleigh distributed, which is a generally accepted assumption for ocean waves at this

time, the relation between the significant

and maximum responses for a linear system is approximately:

X XZ

p(x) = - exp(- ) Rayleigh distribution

m0 2m0

Xal/3 2Jm0

Xal/l000 4Jm

Beukelman (3 used these criteria for an ex-tensive optimisation study for a new patrol boat on the North Sea and Dutch coastel

wa-ters, to be commissioned by the Dutch

Governxrient.

Using the available scatter diagrams of the area under consideration, in which .the rela-tion between significant wave height, peak period of' the spectrum and percentage of occurrence is given, he tried to optimise a given design concept with, respect to opera-bility. For the motion calculation he used a linear strip.theory.mathematicàl model for the calculation of the. heave and pitch

motions. of. the ship as developed by the

Deif t Shiphydromechanics. Laboratory. Within the constraints imposed' by the desïgners, he generated an systematic series of design variations with respect to length, beam and draft. Forward speed was considered to be

constant.

For all the wave spectra in the scatter diagrams and all the different designs he

calculated the significant values of the heave and pitch motions and the vertical accelerations at the centre of gravity of

the ship and at the bow. By introducing, the limiting criteria for the vertical accelera-tions, as outlined, he was able to calculate the operability of the ships.

In the scope of the present paper only the

change in beam of the parent 'design is

considered. A typical result as derived by Beukelrnan for the dependency of the opera-bility on the beam of the vessel is given in Figure 14. The main particulars of the three desi9n variations are presented in Table 6. Beukelman concluded from his. calculations

that increasing the beam of the vessel resulted in an increase of operablity, as demostrated in the figure.

X i oe -kc1n.n FIshIp SI9AII. 5 Fastshìp Maxim. w 4 6 40 5 5.2 5.4 ,56 58 6 62 6.4 6.6 6 8 BEAM

Figure 14. Operability of planing hull with varying beam.

Table 6

He 'himself remarked already in his conclu-sions', that, since non linear 'effects were not taken into account, some care had to'be taken by interpreting these results. Because increasing the beam by some I0 while

keep-ing the draft constant inevitably meant

reducing the deadrise of 'the ship.. Reducing the beam had the reverse effect on the dead-rise. The deadrise varied between 22 and 28 degrees for t'he particular designs. The effect of this was not, taken into account in the linear calculation routine he usèd except by a marginal change in added mass and damping.

The calculation of t'he motions and vertical accelerations with the same designs in the same wave spectra have been performed using

the non linear model as described in the

present study. The respective reference

positions of the three designs with respect to trim and' sinkage arepresented' in Table 6, together with the calculated'values for Ka and a.

Table 7

Due to the fact that the design speed of the vessel is not realy high, i.e. volumetric Froude number around 2.0, the rise of the centre of gravity is rather low.

length ' m 25.00 25.00 25.00: beam rn 6.04 5.42 4.801 displacement m3 93.20 '83.60 74.O0 .deadrise degr. 22 25 27 Is/B _4.34 _4.84 5.49i A/V2/3 6.28 6,.06 5.80i Fnv 1.93 1.96' 2,.00 O degr. ' 2.88 2.45 2.20 z m 0.03 0.01 0.00 ka , a 1.204 0.69 1.258 0.74 ' 1.281 ' 0.77 Fn7 1.93 1.96 2.00 avl/3 0.50 * g acql/3 0.35 * g 60' A 6 o 2Q

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Trim and sinkage however show a trend over the various designs in accordance with real life experience: the wider vessel has more lift and trim angle compared to the narrow one. The generated simulation of the motion; in the time domain had a duration which al-lowed an °encounter" by the ship of atleast 250 waves in each spectrum to allow suf fi-cient statistical accuracy of the derived results.

The results have been elaborated with the

useof the scatter diagrams in two different ways.

First the same criteria with respect. to sig-nificant values for the vertical accelera-tions have been used. The resulta of these calculations are presented in the figure with the asterix, to gether with the results found by Beukelman. it is obvious that using the non linear theory the trend with respect to the béam is reversed::, increasing the beam reduces the operability of the vessel, in this respect it should be noted however that due to the design variation chosen by Beu-kelman there is quite a considerable change in displacement. This might mask the trend even more, because the heavier (beamy) ves-sel will due to its weight experience

some-what lower accelerations than the lighter

(narrow) vessel.

Secondly it has been assumed that the crite-ria formulated ref ering to the significant values of the vertical acçelerations are

actually derived from the hypothesis that

the maximum peak values encouterd are twice the. significant values. This implies that the occurence of a vertical acceleration of 0.7 g at midship or 1.0 g at the bow are the actual limits for operability.

When these criteria are used to calculate

the operability of the vessels, using the suggested non linear approach, the operabi-lity is reduced with another l5 when

compared to the calculations based on the

significant values of the accelerations. The trend of both calculations based on the non linear model however remain the same..

CONCLUSIONS

The non Ïinear mathematical model described in this paper appears to be an adequate tool to calculate the heave and pitch motions of fast ships in head seas.

Although

certain simplifications are inevitable been made,

the inclusion of significant non linear

effect is of prime importance when extreme motions of these craft are to be predicted. For operability calculations in a seaway the use of linear theories may produce both op-posite trends when optimising hu'l parame-ters as too high values of operability when extreme motions are .not adequatly predicted.

REFERENCES

(I J Blok, J.J. and Beukelman, W.

The high speed dsiplacement ship sys-thematic series hull f orrns.

SNANE Trans., Vol.92, 1984., pp 125-150

[.2 ] Beukelman, W.

Prediction of operablility of fast semi planing, vessels in a seaway.

Report 700, Shiphydromechanics Labora-torium TU Deif t, January 1986

(3 J Beukelman, .

Semi planerende vaartuigen in zeegang, predictie inzetbaarheid.

Report 658.-O, Shiphydromechanics Labo-ratory TU Deift, March 1985.

(4 J Bosch, J.J. van den

Tests with two planing boat models in waves.

Report 266, Shiphydromechanics Labora-tory TU Delft, 1970

(5 J Clement, E.P. and Blount, L.D.

Resistance teats of a systhematic

series of planing hull forms. SNANE, 1963.

(6. J Hwang Jong Heul

Added mass of two dimensional cilinders

with the sections of straight frames

oscillating vertically in a free:

surface.

Journal Society of Naval Architects

Korea, Vol. 5, No. 2., 1968 [7 J Faltinsen, O. and. Zhao, R.

Numerical predictions of ship motions at high forward speed.

Phil. Transactions Royal Society London. (1991) 334, pp 241-252.

[8 J Gerritsma, J. and Beukelman, W

The effects of beam on the hydromecha-nic characteristics of ship hulls. 10th ONR Symposium, June 1974., Boston. [9 ] Kant, R.

Tijdsdomein simulatie. programma voor de bewegingen van schepen in onregelmatige golven (1989)

Student Thesis Delft University of. Technology.

[lo] Keuning, J.A. and Gerritsma, J.

Resistance tests with a series of

planing hull forms with 25 degrees deadrise.

International Shipbuilding Progress,

Vol.

198.

[11.] Keuning, J.A.

Distribution of added mass and damping. along the length of a ship model moving. at high forward speed.

International Shipbuilding Progress.,

Vol. 410, pp 123-150. [12] Keuning, J.A.

Invloed van de deadrjse. op het zee-gangsgedrag van planerende. schepen. Repor.t 794-O, ShjJphydromechanics Labo-ratory TU Delft, July 1988.

Keuning, J.A.

Non linear mathematical model for the heaving and pitching of planing boats

in irregular waves.

Shiphydromechan.ics Laboratory TU Delf.t, 1992.

Kring., D..

Investigation of the Zarnick non linear model of planing craft motions.

Report 786, Shiphydromechanics Labora-tory TU Deif t, February 1988.

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-[15] Tasai, F.

On the damping force and added mass of ships heaving and pitching.

Reports of Research. Institute for Applied Mechanics, Kyushu University, Japan, 1960.

Savitsky, D.

Hydrodynamic design of planing hulls. Marine Technology, October 1964. Savitsky, D. and Brown, P.W.

Procedures of hydrodynamic evaluation of planing hulls..

Marine technology, Vol. 13, No,. 4, October 1976.

[18.] Shuford, Charles L.

A theoretical and experimental study of planing surfaces including effects of cross section and plan form.

Report 1355, NACA, 1957. (19] tJrsell, F.

On the virtual mass and damping of floating bodies at zero speed aheacL

Proceedings Symposium Behaviour of Ships in a Seaway NS Wageningen, The Netherlands, 1957.

(.20] Zarnick, E.E.

A non

linear mathematical model of

motions of planing boats in waves. AIAA, 1979.

Non linear heave and pitch motions of fast ships in irregular head seas