TECHNISCHE HOGESCHOOL DELFT
AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDELABORATORIUM VOOR SCHEEPSHYDROMECHANICA
Rapport No. 49P
HYDRODYNAMIC COEFFICIENTS OF RECTANGULAR BARGES
IN SHALLOW WATER.
Second International Conference on
Behaviour of Off-Shore Structures
BOSS'79
.J.A.Keuning and W.Beukeiman
august 1979
Delft University of Technology
Ship Hydromechanics Laboratory
Mekelweg 2 Delft 2208
Second International Conference on Behaviour of Off-Shore Structures Held at: Imperial College, London, England
28to 31 August 1979
HYDRODYNAMIC COEFFICIENTS O RECTANGULAR BBGES I
SHALLOW WATER
J.A. Keuning
Deift Hydraulics Laboratory, The Netherlands
and W. Beukelman.
Deift University of Technology, The Netherlands
Summary
Forced oscillation tests In vertical and horizontal direction have been carried out with
a
2m model of a rectangular offshore pontoon and a cutter dredge pontoon on shallow water.
Scale ratio i : 32.5. The waterdepths were 4.5 i.75 and 1.2 times the draft of the model.
The frequency ranged from 0.2 Hz up to 2.0 Hz. Moreover the amplitude of oscillation
was systematically varied to obtain a check on linearity.
All experimental results arepresented In Figures tO show the influence of water-depth,
amplitude and frequency on added mass and damping.
Some of these experimental results have also been compared with corresponding
cal-culated values.
The calculation methods used are based on strip theory and a diffraction model.
Sponsoredby: DeIftUnlversItyof Technology The Netherlands
Massachusetts Institute of Technology. U.S.A
TheNorweglan Instituteof Technology.Norway University of London. England
Secretariat providediby: BHRA FluId Engineering Copyright: © BHRA.Fluid Engineering
Cranfield. Bedford England
Nomenclature.
Av wave amplitude ratio a,b,c,d,e,g hydrodynamic coefficients
a added mass, transformation coefficient,, subscript for amplitude.
B beam
b damping coefficient
C,
dimensionless sectional added mass2 force exerted by oscillator rod
F coefficient for influence of waterdepth
nh
-vr
reál part of the vertical hydrodynamic force :
imaginair part of the vertical hydrodynamic force
g acceleration due to gravity
h . waterdepth
I mass moment of inertia of pontoon ôr cutter
k wave number for deep water
k0 wave number
L length of pontoon or cutter
1 distance between oscillator legs N number of transformation coefficients
T draft of pontoon
t time :
1 yertical veldcity
xi .
y right hand coordinate system . V
x surge
y sway displacement
z heave J
c phase angle between force and motion
Ca wave amplitude e pitch angle X wave length p density of water roll angle yaw angle
w circular frequency of oscillation
V volume of the displacement of pontoon
1. Introdûction.
For the calculation of ship motions in waves with the commonly used methods an
accurateknowled'ge of the hydrodynamic coefficients, i.e. added mass and damping, as function of the frequency is' essential.
For average ship forms in deep' not restricted water these coefficients are fairly well-known and reliable calculation methods do exist. This is not the case for some
situations which are nowadays more frequently met and in which the motions of floating constructions are important. These are:
- rectangular pontoons or barges, anchored or moving at slow speed,
- ships' at shallow, water, i.e. moored ships and ships approaching harbours with small under keel clearance,
- cutter dredgers working in exposed areas.
All these situations cause special problems with respect to the forementioned calcu-lation methods and large discrepancies between the different methods occur:
- pontoons gie rise to problems due to both the rectangular shape and the large beàm to draft ratio (B/T),
- shallow water effects can until now only be calculated with a limited number of methods especially for the case of small waterdepth to draft ratio (hIT); they are mathematically complex and' lack extensive experimental verification,
- cutter dredgers pose a special problem due to the presence of a slot in the
pontoon body,, which makes it partly a double 'hull ship.
Experimental values for the hydrodynamic coefficients in all these situations are scarce.;
There-fore the Delfr 'Hydraulics Laboratory and the Delft University of Technology decided on a large experimental project to gain sme insight into these problems and to obtain a set of reliable experimental results.
-For this reason forced oscillation tests- and wave force measurement tests with both a pontoon model and a- cutterd'redge pontoon model in three different
water-depths h-ave been carried out.
In this paper the experimental values for added mass and damping are presented and compared with the results of some currently used cakulation methods.
1.1. 'Review.
-For deep water the calculation methods to determine the hydrodynamic coefficients
are well-known '(Ref. 1,2,3,4,) and mainly based- on the work of Ursell (Ref. 5) for
-oscillating-cylinders in a free surface.
The greater part o-f the experimental verification has been performed for normal ship forms by forced oscillation mod-el test-s (Ref. 6,7) and only a small part was related to cylinders with rectangular cross-sections. For this case extensive experiments have been carried out by Vug'ts (Ref. 7,8) with respect to heaving,
-swaying and- rolling motions and also by Takaki (Re-f. 15). -
-Generally the experimental results show a good agreement with the calculations except for the rolling motion where viscous effects play a dominant role
For the case of finite waterdepth several calculation procedures have also been
developed to -determine the- hydrodynamic coefficients. It is worthwhile to mention
in this -respect the- -work- o-f Porter (Ref. 1),, Kim (Ref,. 9), Keil (Ref. 10,11-),, Van Oortmerssen (Ref. 1-2) and' many othe-rs (Ref. I3.14.15.16.17.1'8.-19,,24,26).,
A special 'method to determine the sway added-mass coefficients for rectangular
sections i-n s-hallow water has been- presen-ted b-y F'lagg and Ñewmán in -(Ref. 20).
An important conclusion from the calculated: results is that for a water depth/ draf t ratio smaller than 4, the hydrodynamic coefficients gradUally increase with respect to the values for infinite water depth, while for a water depth! draft rati,o o-f' 2 the changes- a-re very substantial-.
Very few -experiments on -shallow water have been carri-ed out- so far.
Forced oscillation- -tests- with a ship mode-1 in shallow water h-ave been conducted by Tasai (Re-f. 21), while mot-ion- measUrements f,or sway, yaw and roll have been performed by F'ujii and Takahashi '(Ref. 22-).
The inflUence -of shallow water on -hea\e and pitch has been tested by F-reakes and Keay (Ref. 23) for a ship model, while i,t
Tan in (Ref. 27),.
Experiments with circular cylinders in shallow water have been carried out by Yu and Ursell (Ref. 24) to determine the damping by measuring the wave amplitude
ratio.
Up to now no forced oscillation tests in shallow water. are known for determining the hydrodynamic coefficients of cylinders with rectangular sections and a minimum beam/draft ratio of about 4. Thesè values of the beam/draft ratio are often
exceedéd in the case of.pontoons. .
2
Eeriments.
.The experirnefits have been carried out in a bäsin of the Deift Hydraulics Laboratory.
The dimensions of. the basin are: length 40 m, breadth 35 in and' maximum attainable waterdepth 0.70 m. On each side of the basin wave absorbing beaches made of gravel with a slope of-1:1O were construced which guaranteed minimal reflection of the
generated waves. The bottom of the basin has been extensively smoothed and made horizontal within small tolerances.
In the middle ofthis basin a frame with cylindrical piles of small diameter suppor-ted the mechanical oscillators. The supporting frame could be adjussuppor-ted in height to suit the different waterdeptbs. The resonance frequencies of the frame were all very large compared with the, forced oscillation frequencies as' used during the experiments. For the test two oscillators o'f the Planar Motion Mechanism type were used': one for the vertical motions i.e. heave, pitch and roll and one for the horizontal motions i.e. sway, yawand surge. Two models have been used: one of a rectangular pontoon and one of a cutter-dredge-pontoon. Both models were const'ructed of aluminium and were identical except for the slot in the cutter dredge pontoon. The two models 'had sharp edges. For the main particwlars reference is made to
Figure 1.
Due 'to constructional problems it was not possible 'to fix the' axis of rotation during the pitch and the roll modes through the centre of gravity of the models;
consequently some surge and sway motion was present during the'se tests.
The models were connected to the oscillators by means of strain gauge dynamometers, measuring the forces in the direction of motion. These forces were reduced into an
in phase component and a 90 degrees out-of-phase component with the motion, by means of an anàlogue Fourier Transformer, mechanically linked with the oscillators as described in (Ref. 6).
With these force components and the known particulars of the models the added' mass and damping could be calculated using the formulae given in Appendix 1.
With the two models a series of tests' has been performed 'using 12 oscillation
frequencies in the r ange from '0,2 'Hz upto 2.0 Hz wit'h three' amplitudes fOr a
check on the linearity; and with three different waterdepths, i.e. 4.5, 1.75 and 1.2 times the draft of the model to investigate the influence of the waterdepth on added'
mass and damping. '
-3. Calculations.
The forced oscillation experiments with the model pontoons at a waterdepth of 0.5 rn are considered to have no influence from the bottom; For this reason calculations assuming infinite waterdepth 'have been carried out.
After the determination o-f the sectional ad'ded mass and damping use was made
of the well-known strip theory as presented in (Ref. 6) and by integration o-f the two dimensional cross sectional values over the length of the pontoon, results have been obtained for the pontoon.
The hydrodynamic coefficients with respect to deep water have been' determined in'
two ways viz.
1. theciose-fit mapping method as dêscribed by de Jong in ('Ref. 3) and based on Ursell's solution as presented' in (Ref.' 5) for the problem of an infinitely long cylinder oscillating in the free surface.
-With the aid of a conformal transformation df the pontoon-section' to the 'unit
circl'ê the hydrodynainic forces could be computed' an'd the results are shown in
Fig. 6-7 for heave, pitch, sway and yaw 'n a dimensionless" form as denoted in Appendix 1.
The transformation formulae whi'ch were used, are given by:' ' '
z = a n=O 2n-1
N
-(2n--I)
(1)
where: N. = 2 for the three coefficient Lews transformation
N = 5 for the case considered.
.2. the Frank close-fit method, (Ref.4) uses a distribution of pulsating sources along the contour of the two dimensional cross-section.
.
This method is based upon the determination of the velocity potential obtained for a distribution of source singularities over the submerged section. The sources which satisfy the linearized free-surface condidon and the kinematic boundary condition on the surface of the seètion-clinde.r have to be infinite in number to obtain a continuous source distribution. Frank came to an approxi-mation by using a finite source distribution in süch a way that a constant
strength was considered for some straight segments replacing the original section. The hydro.dynamic pressures are obtained from the potential by means of the.
linearized Bernoulli 'equation. Integration of these pressures over the submerged
section cylinder gives the hyd'rodynami:c forces.
The accuracy of this solution fOr normal ship forms depends on the chosen number of source segments. For the pontoon-section considered a number of 12
segments was used.
The hydrodynamic forces represented as added mass and damping have been
caIcu-la'ted with a computer-program described and: presented in (Ref.. 4).
The results are shown in Fig. 6-7 with respect to the same motions and presented in the same way as denoted for the forementioned close-fit mapping method.
For shallow water Keil developed 'a method in (Ref. iO,,fl) to determine the sectional added mass and damping for both vertical and hoizontal motions.
Porter first indicated in (.Ref. I) a way. to find the ve'Ioc:ity potential for
'a cylindrical body performing harmonie motiOns in the free surface of a fluid with finite depth. This method was déveloped for circular cylinders by
Yu-Ursell in ('Ref.. 24,) and for Lewis-form sections by Kim in (Ref. 9).
Keil extended this work and made also use of Lewis-transformation according to the indication N2 in eicpression (1).
A velocity potential was synthesized from appropriate functions satisfying the
boundry conditions on the free surface, the bottom and the body-surface while
delivering radiated waves far f,rom the 'body. Afterwards the pressure distribution
was determined and integration of these pressures' over the body-.surfa'cé gave the
hydrodynamic forces in a fluid of infinite depth.
As input for the computerprogram a set of wave-lengths is used from which the
frequency is 'given by:
w
\Jkg
thg (kh) (2.)in which 'k = = wave number
:0 X
X =' wave length
The. real par.t of the. vertical hydrodynamic force may be written as the product of sectional, added mass and acceleration and is written' as
E = a' a
Vr zz (3)'
with.: a' sectional added' mass
.zz
a the vertical acceleration
The sec.tional added' mass for the vertical motions is e.xpressedr.in a dimensionless way with:' 'a' zz = p B'2 (.4) in which:
B = breadth of the section at the waterline.
The imaginary part of the vertical hydrodynamic force is equal to the product of the sectional damping and the velocity and may be written as:
Vj zz (5)
in which: b' = the sectional damping V = the vertical velocity
This force may also be exoressed in terms of the squared amplitudé ratio of the radiated waves and the motion by:
FVj = A2pti) v
k2Fh
(6)in which:
with:
a amplitude of radiated wave z = vertical motion amplitude
¿ 2
.
k = - = wave-number for deep water
. g
-F = cosh2(k0h) (7
nh k h + sinh(k h) cosh(k h)
o o o .
From (5) and (6) follows the sectional damping coefficient:
b' = PWk2 (8)
Analogous expressions are derived by Keil in (Ref 11) for the horizontal motions in shallow water. The results determined with the computer-program according to Keil are shown in Fig. 6-7 for heave, pitch, sway and yaw, related to the three water-deoths considered This presentation is also in a non-dimensional form
For the cutter-dredge pontoon no calculations have been carried out up to. now, but it seems reasonable to perform this. in the futureby considering a part of the pontoon as a catamaran.
To solve this problem for the deep water condition reference 'should be made to the work of De Jong in (Ref. 25), where he presen.ts a method of determining the hy.dro-dynamic coefficients of two parallel identicàl cylinders oscillating in the free sur f ace.
In all calculations the rolling coefficients have been, left out of' consideration because comparison with the measurements is not possible due to the fact that the centre of rotation was not situated at the centre of gravity. Corrections should be necessary for the position of the centre of gravity and to eliminate the para-sitic swaying motion which took place during the rolling experiments. However the accuracy of these corrections might hardly be sufficient to obtain reliable
final results.
Van Oortmerssen (Ref. 12) formulated a method in which the three dimensional character of the flow around a ship on water of restricted depth can be taken into account. The method is based on. linear potentional. theory and supposes therefore an idea-1 fluid
and small amplitudes for both the waves and the motions and uses a three dimensional source technique. With aid of the linearized free surface condition and the boundary conditions on the sea floor, the ship's surface and at infinity the velocity poten-tial is found. The pressure on the ship's surface is than calculated .using Ber.nouilis theorem. Integration over the ship's surface yields the hydrodynamic forces. For the calculations an average number of ten sources over one wave length has been used. Results proved to be consistent for changes in the source distribution. Due to lack of time and the high costs involved 'only a limited number of calculations has been
made.
The calculations were made for the pontoon on deep water and for the cutter-dredge pontoon on the waterdepth of 1.75 times T.
The results of these calculations are presented in the Figs. 6-7 in a d.imensiionÏess form, as described: in Appendix I:.
4. of the results,
4.1. Experimental results.
The results of the tests are presented in a dimensionless form by using the formulae given in Appendix 1.
The added mass and damping for all motion còmponents for both the pontoon and the cutter dredge pontoon are presented in Fig. 2-3 and on basis of the dimensionless
frequency parameter:,
-j
'for one amplitude of oscillation and the three different waterdepths.
In Fig. 4 the dependency of added mass and damping on waterdepth :5 given for the most inìportatt motion components for three different frequencies. In Fig. 5 the dependency of the same hydrodynamic coefficients on the oscillation amplitude is shown for the three different waterdepths investigated and for the considered three frequencies of oscillation.
These last two figures are based on the results of the test with the pontoon only. For the cutter-dredge pontoon the figures are the same in a qualitative way,
From these figures it can be seen that for the heave motion -added mass increases considerably with decreasing wa-terdeoth', giving the steepest increase for water-depth to draft ratio!s smaller than 2.
For large frequencies the added mass tends to become constant as shown in Fig. 2. The dependency of 'the added mass on the amplitude of oscillatiion is small for a'll three waterdepths investigated. The damping curve tends to zero for both very large and very small frequencies for the deep water situation only. With decreasing warerdepth the damping increases with the increase of os-ciliation frequency. This
increase is most evident fdr -the smallest waterdepth. The damping is strongly d-pendent on the amplitude of oscillation, as -shown in Fig. 5
Over the whole frequency range the added mass and damping for the cutter dredge pon-toon is less than for the ponpon-toon. This difference increases with the decrease of waterdepth.
The same qualitative tendencies can be seen for the hydrodynamic coefficients in the case of the pitch motion.
For the sway motion the influence of the waterdepth on addéd mass and damping in-creases'with the decrease of theoscillation frequency. Especially for the low frequencie.s the influence of waterdepth on added mass i-s ignificant. For the -dam-ping this tendency is much less pronounced. These dam-dam-ping curves however, show only a small tendency to become zero for even the highest frequencies- investigated,
although small damping was t-o be -expected for much lower frequencies.
Obviously viscous effects play an important role- at the higher frequencies. At these frequencies a growing amount of eddy shedding from the sharp corners- could be ob-served-. Nevertheless both added mass and' damping remain remarkably independent on-the oscillation amplitude, see Fig. 5. In general on-the pontoon has less addèd mass than the cutter dredge pontoon contrary to the vertical motion.
For the damping these differences are fairly small-. As fa-r as sugi-ng'is-- concerned,
the influence o-f the waterdepth is generally small on- both added mass and damping, although the latter tends to increase slightly for the smallest -waterdepth. The
in-dependenceon bo-th added mass and damping on the oscillation- amplitude-
-is fair for the lower frequencies but tends to decrease with ±nreasi-ng frequency.
Here also the damping increases wit-li the increase of--oscillation frequency, and no tendency to decrease to. zero damping fo-r high frequencies can be observed. This
ten-dency is equal for all th-ree waterdepths investigated and appears to be 'even more
pronounced than' for swaying. Tn -general the differences between the. pon-toon and the
cutter dredge pontoon are- small. For yawing the 'hydrodynamic coefficients show the same tendencies as- for swaying. In analysing the results of the roll tests it
should be emphasised t-ha-t during thèse tests -the a-x-i-s of rotation- did no-t go
the moment of inertia of the oscillated models,, that the model performed a- simul-taneous roIl and sway motion. A correction method applied to the results to eliminate the effects of this sway motion on the forces measured gave no satisfactory results especially for the lowest waterdeoth. Therefore the original results are presented in Figs. 2 and 3. From these figures it can be seen that the hydrodynainic coefficients for roll show little dependency on waterdepth. Here also the damping does. not tend to zero for higher frequencies. Both added mass and damping show an evident depen-dency on the oscillation amplitude.
i-n general the pontoon has less added mass and damping compared with the cut ter dredge pontoon.
4.2. Comparison of experiments and calculations.
FOr the deep-water situation it appears from Figs. 6-7 that there is a good agreement between the results of both calculation methods considered and the experiments. The differences between the results of the fit mapping method and the Frank close-fit method appeared to be rather small.
For the higher frequencies the prediction of the damping coefficient according to both mentioned calculation methods is somewhat too low which might be due to viscous effects which might increase with the frequency.
The results of the calculation method of Keil showed for 'deep water over the greater part of the frequency rañge good agreement with both other calculation methods and also with the experimental results. For -the low frequencies, however, there
appeared to be a rather strong deviation from the experiments and the other calcu-lated results. This holds especially for the vertical damping coefficients.
For the second wa-terdepth it appears from Fig. 6-7 that the prediction according to Keil's method is rather satisfactory for the added mass except for the lower fre-quency range.
The agreement for damping, however, is rather poor especially for the low frequencies where there is an inexplicable tendency 'to infinity.
For the smallest waterdepth the deviations from the experiments are even greater particular for the damping coefficients of the vertical motions.
it is remarkable that with respect to the calculated results according to Keil's method the strong increase of damping occurs at higher frequencies when the water-depth decreases.
It aooears from Fig. 6 and 7 that the calculations according to the method of Van Oortmerssen give both for deep water and shallow water satisfactory results. For shallow water only the second waterdepth for the cutter dredge pontoon, has been considered, It is an advantage that this method is able to take into account the "twin hull" part of the pontoon. The discrepancies are only significant at the highest frequencies, especially in the case of added mass for the surging motion. However, it should be kept in mind that it is to be expected tha for these high frequencies viscous effects grow more important.
5-. Conclusions and recommendations.
From the oreceeding experiments and calculations the following concl'úsions and
re-commendations may 'be derived with respect to the hydro-dyn'amic coefficients 'of a
pontoon in shallow water:
The influence of waterdepth on added mass and damping 'is -most important for the vertical motions. Both added mass and damping increase with decreasing waterdepth.
The dependency of added mass and damping on the oscillation
ampli-tude is more- evident for the vertical motions than for the horizontal motions,, while the dependency increases with decreasing waterdeoth.
-3,. The damning appears to increase with frequency for the vertical motions at the
lowest waterdept-h h/T 1.2) and' increases with f req,uency for surging a-t ail
waterdepths.
4. The cutter dredge pontoon has lower added mass and damping than the pontoon for the vertical motions. For the horizontal motions the opposite statement holds true.
5 The calculated values f added mass and damping for deep water agree very we1l with the experimental values for the highest waterdepth considered th/T 4.5).
The calculated results with respect to shallow water according to the method of Van Oortmerssen show a good agreement with the experimental values, while those
according to Keil's method needs improvement especially for damping
The experimental values for added mass and damping of the rolling motion needs further analysis for a good comparison with the calculated results This compari-son with resnect to damping, however, will remain difficult on account of the dominant viscous influence for the rolling motion.
6. Acknowledgement s.
Special thanks are due to the various members of the staff of the Delf.t Hydraulics Laboratory and Ship Hydromechanics Laboratory for their assistance in running the
7. References.
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cylinders oscillating in a fre surfäce." University of California., Inst.. of
Eng. Res:., Berkelçy. (July 1960).
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Jong,, B. de: "Computation of the hydrodynamic coefficients of oscillating
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(November1969).
Frank, W. and Salvesen. N.: "The Frank Close-fit Ship Motions CompüterProgram."
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fluid." Ouart. J. of Mech. and Appl. Math., Vol. 2, pp. 218-231 (1949).
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Ii,
no. 123, ppi 506-522 (November 1964).
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i i2S,
(May 1968).
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Bewegung zwei-dimensionaler K6.rper an der Oberflche flacher Wasser). Institut
f'tr Schiffbau, Bericht Nr. 305,(February 1974).
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F61,(19.76).
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16
Kan, M.: "The added mass coefficient of a cylinder oscillating in shallow water
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and
k_.4..co." Ship Res. Inst., Japan, no. 52,(1977).
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....
Sayer, P., and Ursell,. F..:: "On 'the vir'tuaF,mass,, at. 'long'
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a yy b yy a = xx b = xx
xw
a=.z
a00 YaW { aThe equations of motion for forced osci1lation in the six modes are given by:
surge
a(pV+a
k (F+F.)sin(wt+ e)
xx xx 1 2 x sway (pV+ a) + b
4-d+ e
= (F1 i-. F2)sin(wt + e) heave(p7
+ a),
+cz +
+ e0Ö + = (F1 + F2)sin(wt + e,) (A3)roll
-xx + + + c = (F1 . F)
-gin(wt ± e.) (A4)
F) - sin(wt +. e0)
(A5)
yaw
(1
+ a) + b,1' +
+ = (F1 - F2) sinWt +(A6) Substitutng the known motion f the mödel and its deivat ives and reducing the measured fórces into domponent in phase and one 90 degrees out of phase with the motion it can be shown that:
=
--_
{(1 4-
F2)cos e + - pV b =._L J(F
+ F )sine ZZZaW L1
2 z ((F1 + F2)cose - pV + F2)sine + F ):sine 2 F2)-F)
2 cose PV X ji
-- cose 1 2 b00 = 'e-E [(F1 - F2) Sine01
116 cose+ C
O - I OOOa
yy = - F2) -SflC
+Cq
-(Al)
(A2).a
{
1.
. s.inE:.
For the rotations pitch and yaw:
added mass moment of
inertia divided: by the mass of the. modis times the length squared
damping . : divided' by the mass of the models times the length squared and multtpiied with the square root of half the breadth divided by the gTavitational aceleration:
*
\B/2g
For Ùhe rotation roll : the same as for pitch and .yaw. except for .L2 in. both terms
which has been replaced by the breadth of the models
squared i.e,.: and * i I
-COSE:
i-I
2 p zz j!The cross coupling coefficients are omitted here because. theeernta1 results
need further. analysis to be presented. .. .
in the .prsentation of the results the added mass and damping coefficients have been made dimensionless using the following formulae:
For the translations heave, surge, sway:
added mass : divided by mass of the models
damping : divided by the. mass1 of the models, pV and multiplied with
the square roo.t of half the breadth divided by jravitationa.l acceleration: . .
o
D
1:0.11
2m.
CUTTER DREDGE PONTOON
0.60
118
PONTOON
0.18
Fig. I Used barge models
EXPER I MENTS
-
PONTOON
CUTTER DREDGE
PONTOON
D h/T= 1.20
o
=1.75
L
=455
Fig. 2 Added mass related to frequency of oscillation for six modes of motion
41 u 41
-:
r
W -. W II r, 425 swa 455 4' 5 25 W 4W -455 4W 415 410 24 50 45 I,--1 w: Cil-EXPERIMENTS
PONTOON
CUTTER DREDGE
PONTOON
D h/T= 1.20
O
= 1.75
L:
=4.55
Fig. 3 Damping related to frequency of osci11ation for six modes of motion
120 1
ir
. 40 I5 ISO VS 0 44 445 L 445 LL4
4aP
ISO H 40 450 - 0M VI 01101 - . Wig 45Vo
'---
--çS.
N
w,
2g =
0.5---= tO
=1.5
Fig. 4 Added mass and damp.iitg related to watrdepth draft-ratio for the póntoon
122
"w
Fig. 5 Added mas anddamping related' to amplitude of oscLilation for the pontoon
D.
h/T = i.2O..
=1.75 O
EXPER I MENTS
PONTOON
CUTTER DREDGE
PONTOON
CALCULATIONS
D h/T= 1.20
O
= 1.75
KEIL
= 4.55
.
. . CLOSE FI T MAPPING
. . .. FRANK CLOSE FIT
J van
CUTTER =
PONTOON =Fig. 6 Measured and calculated added mass related to frequency of oscillation 4. u' J u I q.
s
X--/
:-::
o --fi'
-4 USASE 4U 4' Q,t:
-. 'S S» _5 U o." wo 40' o us SAW 450 4'S loo 525 ISO. .
a/ipEXPERIMENTS
---PONTOON
CUTTER DREDGE
PONI OON
124
Fig. 7 Measured and calcuPated damping related to frequency of oscillation
CALCULATIONS
Is\\
5'... 5'....:
420 430 423 lOO 325 ISO 1'O
4.5 400 4$ 420 -f 430 - o 410 ..---
