R E P O R T N o . 186S (SH/236, 236a, 236b) M a r c h 1974
NEDERLANDS SCHEEPSSTUDIECENTRUM T N O
N E T H E R L A N D S S H I P R E S E A R C H C E N T R E T N O S H I P B U I L D I N G D E P A R T M E N T L E E G H W A T E R S T R A A T 5, D E L F TWAVES INDUCED MOTIONS AND D R I F T
FORCES ON A F L O A T I N G STRUCTURE
( D O O R G O L V E N O P G E W E K T E B E W E G I N G E N E N D R I F T K R A C H T E N V O O R . E É N : D R I J V E N D E C O N S T R U C T I E )
by
I R . R . W A H A B
(Institute T N O for M e c h a n i c a l Constructions)
TO
V O O R W O O R D P R E F A C E
T e n tijde van het uitvoeren van de modelexperimenten aan het booreiland „ N o r r i g " (waarvoor verwezen wordt naar rapport no. 156 S van het Nederlands Scheepsstudiecentrum T N O , [6]), is een onderzoek begonnen naar een mathematisch model waar-mede de bewegingen van en de krachten op booreilanden be-rekend kunnen worden die optreden onder Lnvloed van golven, wind en stroom.
Omdat nog zo weinig bekend was van de laagfrequente drift-krachten voor booreilanden moest het mathematisch model ook geschikt zijn om de driftkrachten te berekenen. Z i j zijn met het oog op de te leveren herstelkrachten van groot belang bij het bepalen van het meest geschikte type voor de hulpmiddelen om een booreiland te positioneren en het dimensioneren ervan.
Omdat geen demping is opgenomen, gelden de door Hooft in zijn proefschrift gehanteerde bewegingsvergelijkingen alleen voor bewegingen buiten het resonantiegebied. De driftkrachten be-staan daarentegen grotendeels uit viskeuze dempingskrachten. Hierdoor is zijn werkwijze minder geschikt voor het bepalen van de driftkrachten.
I n de onderhavige studie zijn de bewegingsvergelijkingen met demping opgesteld. D e dempingscoëfficiënten worden berekend met snelheidsafhankelijke weerstandscoëfïiciënten volgens Hoer-ner. E r wordt een iteratie proces gebruikt om de „ r o o t mean square" waarde van de relatieve snelheid tussen het water en de segmenten van de cilinders te bepalen. Deze relatieve snelheid omvat de absolute snelheid van het segment, de orbitale snelheid van de golf, de massa transportsnelheid in de golfen de stroom-snelheid. Ook zijn de driftkrachten bestudeerd. Hiertoe is het viskeuze deel van de driftkrachten berekend als zijnde de ge-middelde waarde van alle dempingskrachten die op het onder-waterdeel werken.
D e berekeningsresultaten voor de bewegingen die met dit mathematisch model zijn verkregen stemmen goed overeen met de experimenteel op modelschaal bepaalde waarden voor de Norrig.
D e overeenstemming tussen de berekende viskeuze driftkracht en de voor het model uit de gemeten ankerkrachten bepaalde driftkracht is redelijk. Met het oog op een beter inzicht in de geldigheid van de aannamen zijn nauwkeuriger metingen van de driftkrachten gewenst.
At the time of carrying out the model experiments on the drilling rig "Norrig" (see report 156 S of the Netherlands Ship Research Centre T N O , [6]), an investigation was started to make a mathematical model to predict the motions of and the forces acting upon drilling rigs by the action waves, wind and current.
A s until now little has been known about the drift forces in the low frequency range, the mathematical model had to be able to calculate them. In view of the restoring forces that have to be delivered, the drift forces are very important in choosing the most suitable type of appliance for positioning and in giving it the right dimensions.
A s the equations of motion used by Hooft in his dissertation do not include damping effects, they apply to motions outside the range of resonance. The drift forces in contrary are mainly composed of viscous damping forces. In consequence of this the working method of Hooft is less suitable to determine the drift forces.
In the study at issue the damping is included in the equations of motion. The damping coefficients are calculated from speed-dependent drag coefficients by Hoemer. Use is made of an iteration process to determine the root mean square value of the relative velocity between the water and the segments of the tubes. This relative velocity involves the absolute velocity of the segment, the orbital velocity of the water particles due to the presence of the wave, the mass transport velocity in the wave and the current velocity. Furthermore the drift forces were studied. F o r this purpose the viscous part of the drift forces was calculated as being the mean value of ali damping forces acting upon the under water part.
With regard to the motions, the results of calculations with this mathematical model show good agreement with experi-mentally determined values for the model of the Norrig,
The agreement between the calculated viscous drift force and the drift force as determined from the measured anchor forces on the model is rather good. I n order to get a better insight into the validity of the assumptions, more accurate measurements of the drift forces are needed,
T H E NETHERLANDS SHIP RESEARCH C E N T R E TNO
C O N T E N T S
page
List of symbols 4 Summary 7
1 Introduction 7
2 Forces on tubular components exerted by the water 7
2.1 Approach 7 2.2 Oscillating tubular componeats. in waves 8
2.3 Added mass and damping coefficients 10
3 Hydrodynamic forces on footings 10
3.! Approach 10 3.2 Oscillating footings in waves 11
4 Wave induced motions 11 4.1 Calculation o f the motions 11
4.2 Some results obtained with the calculation procedure 12
5 Viscous drift forces 19 5.1 Calculation of the drift forces 19
5.2 Some results obtained with the calculation procedure 20
6 Concluding remarks 21
References 21 Appendix I 22 Appendix I I 23
L I S T O F S Y M B O L S
Co drag coefficient
CM added mass coefficient
c.. function function
D diameter of circular cross section of tube components of drift force
F damping force for total structure hydrodynamic damping forces F total hydrodynamic forces
lx,y,: Froude K r y l o f f force components on segment of tube p
' 2x,ytX
damping force components on segment of tube ^3x,y,T added mass force components on segment of tube
hydrostatic force components hydrostatic moment components F'
x,y,2
force components in Gxj'z-system mass product of inertia
components of total force exerted by the water on a segment of a submerged tube M mass of floating structure
^'x.y.x moment components in G^y^^ system drift moment
P pressure due to wave ^xy function
•^xy function
V volume of footing
V root mean square of relative velocity component prependicular to the tube axis current velocity
v„ mass transport velocity V
' rx,y,z
components of relative oscillatory velocity V
x,yyX
components of total relative velocity V'
x,y,z
components of total relative velocity normal to the tube axis ^(x,y,x)s sine components of Ki,>,j
V(x,y.z)c cosine component of
Vix.y.j) I stationary parts of the total relative velocity components V^ ^ ., W, coefficient vector for tube segment Froude Kryloff wave forces
coefficient vector f o r tube hydrostatic forces a radius of circular cross section of tube
coefficient matrix f o r tube segment added mass and damping forces coefficient matrix f o r tube hydrostatic forces
f depth of cylindrical cross section below water surface 9 acceleration due to gravity
k wave number = InfX
I length of tube
Al length of tube segment
P added mass
q damping
r hydrostatic restoring force
t time
U , V, w water velocity components due to orbital motion in wave
X surge '
y sway
coordinates
a angle of axis of tube with x-y plane
angle of x-axis with the projection of the axis of the tube upon the x--y plane r angle of axis of tube with x-z plane
Ó angle of x-axis with the projection of the axis of the tube upon the wr--z plane R angle of axis of tube with y-z plane
^•x,y,z,tti,e,4i phase lag of motion with respect to wave in G
<!> roll
>! angle between direction of current and x-axis
0 pitch
K angle of y-&\\s with the projection of the axis of the tube upon the y-•z plane
k wave length
/' angle between direction o f advance of waves and jv-axis
'J density of water
w yaw
CO frequency of motion C wave elevation
WAVES INDUCED MOTIONS A N D DRIFT
FORCES ON A FLOATING STRUCTURE
by Ir. R. W A H A B
Summary
A method is given for computing the wave induced motions and drift force of a floating drilHng rig in regular waves of arbitrary direction. The effect of current is included. The drilling rig is assumed to be composed of tubular parts and three-dimensional footings of which the dimensions are small compared with the wave length. The mutual interaction between these components is neglected.
1 Introduction
A free floating vessel i n waves involves two types of motions. The direct wave induced motions have fre-quencies which are directly related to the frequency of encounter of the waves. A t the same time the vessel drifts slowly i n the direction of advance o f the waves. When the vessel is kept in position by mooring lines or propulsors controlled by a dynamic positioning system, the drift motion has an oscillatory character with a period of 1 to 3 minutes. These slowly varying drift motions may be large as compared to the direct wave induced motions and may cause large forces in the mooring lines, see for example [1]. I n this report an attempt will be made to estimate the forces which cause the drift motions of floating drilling rigs with underwater parts which mainly consist of tubular and spherical components.
I t is assumed that the relative motion of a compo-nent of the structure and the surrounding water particles determine the hydrodynamic force on the component. This approach has been successfully used to estimate the forces exerted by waves on piles.
When a floating structure is considered the relative motions result f r o m the motions of the structure, the orbital motions of the waterparticles, the mass trans-port in a progressing wave and the presence of current. I n this set-up the mass transport in a wave gives rise to the wave-induced part of the drift force.
I n order to determine this force, the motions of the structure have to be determined first. I n the subject investigation the wave-induced motions and drift forces are considered f o r the structure floating in deep water and subjected to simple harmonic waves. The effect o f station keeping systems on the motions is neglected, a simplification which is acceptable i n many cases. Another simplification is the neglect of the mutual interference among the components of the structure.
2 Forces on tubular components exerted by the water
2.1 Approach
•Consider first the hydrodynamic force on a submerged cylinder oscillating with small amplitude in the vertical z-direction. Generally this force is expressed as
F, = ^pz-qi i n which
p -- added mass q = damping
Potential theory shows that for bodies dose to the free surface both p and q depend on the frequency of oscillation. However, i f
ƒ > 2a and X>2a
(see fig. 1), the added mass of circular cylinders deviate less than 10% f r o m the value found for an infinitely large ƒ. Moreover, the potential damping appears to
be very small as compared to vdlues, obtained experi-mentally. The oscillating motion of the tube may be of about the same magnitude as the tube diameter or even larger. I n that case the hydrodynamic force appears to be rather non-linear with the motions. For many practical applications the hydrodynamic for.ce is approximated by
Fz = ~pz-q\i\z
Experimental results obtained on deeply submerged cylinders, indicate that p and q do not depend on the frequency o f oscillation, but in some cases they appear to depend strongly on the amplitude of oscillation, [41
A tubular component of a floating structure does
not oscillate in one direction only. In the more general case the two-dimensional circular cylinder given in fig. 1, oscillates along both the y- and z-axes at the same time. The centre of the circular cross-section will then describe the well-known Lissajou patterns. The hydrodynamic forces in the >'-direction may then be expected to be influenced by the motion in the z-direction. Likewise, the force in the z-direction is affected by the motion in the j-direction. Unfortu* nately, to the knowledge of the author, this aspect has not been experimentally investigated yet. The present study is based on the following relationship for the hydrodynamic forces i n the yr and z-directions On a tube of unit length
W,= -^QVDC^y^e'^D'CMy
-igVDC^z-Q^D'C„z
Where V represents the Root Mean Square of the cylinder velocity
r - o o J 0
The above approach is only applicable to cylindrical components with circular cross-section. For the drag coefficient valyes obtained experimentally may be used.
So far only translatory motions perpendicular to the axis o f the tube were discussed. Generally, how-ever, translatory motions take place in an arbitrary direction.
_ I t will be assumed here that ïn this case the resulting hydrodynamic force acts in a direction perpendicular to the axis of the tube.
In fig. 2 a fully submerged tube located i n the x-z plane is considered, having a velocity and acceleration
.z
Fig. 2- Fiilly submerged tubular lelemeht. •.
in the x-direction. The hydrodynamic forces acting o n the tube are then
F^ = ^-^QVDfCoXcose -Q^D^iCj,,x.cossjcoie
F , = ^iQVDlCoX'Cose+Q^D-tCf,,xcosej s i n f
So the force i n the direction of the tube axis is neglect-ed. This approach can easily be generalized f o r a tube with an arbitrary pos'tion in space, oscillating i n the
X-, y- and z-directions. This case is worked out hi
Appendix L
I f a tubular component pierces the water surface, the local added mass and damping of the tube will varv with the distance f r o m the surface.. This effect is neglected in the present approach. This neglect i ^ justified i f the tube length is large as compared t o the. diameter and the frequency o f oscillation isi h\gh
A floating structure has six degrees of freedom. So
'ks tubular components will also execute three trans-latory and three angular motions. These tubular com^ ponents are assumed to have lengths w h i t h are large compared to their diameters. For determining the-hydrodynamic forces these tubes are divided'into small segments: in the present case 10 segments of equal length were considered.
Each segment is assumed to perform translatory motions only. The effect of the angular motions of these segments is neglected. This neglect is not expected tp introduce significant errors i f the segment,? are sufficiently small.
I n addition to the hydrodynamic forces-on the tubu.-lar components hydrostatic forces occur which are generated by the changes of the displacement of the oscillating body when surface piercing parts are present. These forces only act in the vertical directions
2.2 Oscillating tubular' componeiTts in waves
The oscillating part of the hydrodynamic force- is not only determined by the added mass and .dahaping
9
forces due to the absolute motion of the tube segments. In the presence of waves the hydrodynamic force is made up of the following components
- the Froude-Kryloff force caused by the pressure distribution in the undisturbed wave;
- the damping force caused by the relative velocity between the tube segment and the surrounding water. This relative velocity comprises the absolute motions of the segment and the orbital velocity of the water particles at the location of the segment;
- the added mass force caused by the acceleration of the tube segment relative to the water particles. This relative acceleration, involves both the absolute motion of the body and the acceleration of the water pardcles due to the wave orbital motion. When the
x-y plane coincides with the mean free water
surface, the pressure variations at {x^, y^^ Z 3 ) due to deep water waves are given by
P = — QgCae~'"'sinikx^cos fi + ky^ sinn — (Ot)
While the wave elevation at ( X 3 , 3 ^ 3 , 0) is given by
C = Co s i n ( f c x 3 cos n + ky 3 s i n / / - c o / )
See fig. 3 for a definition of the wave direction /t.
The velocity due to the orbital motion of the water particles is in the
direction ol advance of waves current direction
Fig. 3. Definition of direction of waves and current. Since the diameter and length of the tube segments are small as compared to the wave length, the Froude-Kryloff" forces on a segment of a submerged tube are given by
F,,= - ^ ~ - M = ^\nD'MQg{^e-'^^kcosixC,y
F „ = - - | ^ - ^ - A / = +]^D^MQgi^-''"ksmiir
Fu = - l ^ - ^ - ^ ' = -^nD'Alsgt^^e-'-^kS^
Where
C^y = COS(A:A:3 cos^-l-/i:>'3sin/i —cor) S^y = sin(A-A:3COs/z-l-A:j'3 sin/i —0)/)
xy x-direction u = — („cocos^e '^^S, xy xy ^'-direction v = — Cocosin/ue ''^'S z-direction w =—[,a(oe~'"'Cx
The relative velocity becomes
Kx = x-y^ilz+ZjÓ-u V,y = y + X3\p-Z3(i>-v Vrz = é+y3<t>-X3Ö-w
Following appendix I these velocities generate damping forces on the tube segment considered which are given by
f2x = —
g'Fr;cCos^e-l-^K,J,cos}'sin7cos^-l--^-9F„ cos a sin a cos ^
F2y = <]KxCOSBsinEcosK — qV^yCos^y + + qVr^cosasmccsinP
F2~ = ?F,,,cos£sin£sinK-l-9F,j,cos7sin}'sin(5 —
- ^ F ^ c o s ^ a
The relative acceleration is
Kx = X-y3^+Z3Ö-ü Ky=y+X3^-Z3^-V
Kz = z+y3^-x3e-w
The added mass forces become
F^x = —pFrj: cos ^2-1-ƒ» Fry cosy sin y cos
+pVr2 cos a sin a cos)?
Fiy = pKx COSE sine COS K-pV,yCos^y + +pVr. cos a sin a sin
F3z=P Kx cos E sin e sin
K•-^--l-/7 F r y cos V sin V sin «5 - JJ F „ cos
Tubular components which pierce the water surface give rise to hydrostatic forces. Suppose the tube given in fig. 4 to pierce the surface in ( x i , yi, 0). The contri-bution of this tube to the hydrostatic force is
F,x = 0 F,y=0
F(g. 4. Hydrostatic forces due to a tube pierlng the water surface.
Thfe hydrostatic moments amount tp
P4é, = yiF^^-gg l (j) ac\/cps'/? sin^p\ nD
dyj \s\na sin-^a/ 64
A i n V i cos^A oxj V.sin'a sin a / 64
Thus the forces exerted b y the water on a tubular com-ponent are known. These forces are presented in a more accessible way in Appendix I I .
2.3 Added mass and damping coefficients
The added mass coefficient i n the hydrodynamic force follows f r o m
P-= Ö - ^ A / C ^
Data obtaified experimentally or analytically may be used for the non-dimensional added mass coefficient C. I n the present set-up the determination of the damping coefficient involves an iteration process since it depends on the relative motion of the tube segment.
q.= ^VD~AI-CD
For the non-dimensional drag eoefficient Cp data obtained experimentally may be used, f o r example those obtained for cylinders advancing at constant •velocity as given in ,[5].
The velocity V is the Root Mean Square value o f the component perpendicular to the tube axis of the relative velocity between the tuhe segment and: the water.
f h r s relative velocity involves the, absolute motions of the segment,, the orbital velocity of the water-particles due to the presence of waves, the mass transport velocity in waves and the current velocity.
The mass transport velocity in deep water waves iS'
The direction o f V,„ coincides with the direction of advance of waves. The current has a velocity and a direction rj which is defined in fig. 3.
The relative velocities in the .y-, ^-iand z-directions become respectively
F j - = x + ZiÓ—y^tiz — u— y„,,cösp — VcC0S'i]
y, = y + Xjil/-z^(l)~ti-V„,smi:t-V^s\nn
V. = z-\-y34>-XiQ-'iV
The resultant of these velocities has a component along the axis of the tube and a component perpen-dicular to it. The latter may be decomposed again into components along the x-, y- and z-axes.
V'x= Fj.cos'^.E—Fj'Cosy,si'ii7'Cosi5— FjCdsasi'n acosj? Vy = - cos e sin e cos x - | - 1 ^ coi^y — cos a&m» sin /J FJ" = — Fj-cosÊsni-fisln x—Vycosy sin y sln.^M- K,'COs^a
The velocity Pis defined .as
V - ^ = 4 r J ( K . ^ ' + F ; ^ + K'^> d/
3 Hydrodynamic forces on footings 3.1 Approach
The hydrodynamic forces on submerged two-dimen-sional cylinders have been investigated more thoroughly than the forces on distinctly three-dimensional sub-merged bodies. These studies are largely restricted to analytical computations on submerged spheres and ellipsoids. The bodies used as footings of some types of floating structures generally have shapes which are less amenable to an analytical treatment.
I n that case, the use of data obtained experimentally is the only open option. However, in some instance it may be necessary to make an estimate of the hydro-dynamic forces. I n some rare cases it may bé possible to use experimental data as published! Jn some other cases the following notes apply.
The footings of a rig at its deep (operating) draft are generally located well below the water surface: the centre of the footings being located twice-the height o f the footing below the water surface or deeper.
The results of analytical calculations o n spheres indicate that the effect of the water surface on the hydrodynamic forces is small. The added mass may then be assumed to be equal to the value in an u n -bounded fluid,, which has been determined f o r many simple shapes. The potential damping of these rather deeply siibmerged bodies is small. The viscoU|S
damp-ing could be estimated f r o m the drag coefficients of similar bodies at constant velocities, in a manner similar to that used for the tubular components. Since in that approach the damping force is assumed to have a direction opposite to the velocity, this calcula-tion method would essentially only be valid for spheric-ally shaped components. This restriction is rather severe. A more general, yet simple, calculation proce-dure can only be obtained by further simplification. Footings of floating structures, i f present, are usually rather big, their significant dimensions being 10 to 20 m or more. (Tubular components have diameters which range f r o m 2 to 10 m maximum).
The limit o f operation of many floating structures is reached, or even exceeded, when the significant wave height attains a value of 5 m. So the motions of the footings may be expected to be generally small i f "workable conditions" are considered. This leads to the supposition that the non-linear viscous damping is not as predominant as with the tubular components. Experimental data to support this supposition are not available, however.
Although the damping of the footings may be expected to be small, it may not be negligible as pared to the damping caused by all the tubular com-ponents together. A n erroneous estimate of the damp-ing forces due to the footdamp-ings may cause the predicted behaviour to deviate significantly f r o m experimental values; however, this will only be apparent i f the footings are large and if the motions are near resonance.
3.2 Oscillating footings in waves
The hydrodynamic forces exerted on big three-dimensional components are assumed to depend linearly on the reladve motion between the water and the body. The centre o f gravity of the displaced volume of the body is located in {x^, ƒ 4 , Z 4 ) .
The relative velocity is given by
Vrx = x-y^il/ + z j - u Ky = y + xj/-z^(l>-v Kz = z+y^(i>-z^(}-w
The forces exerted by the water are
Py = -VyVry-CiyVry-^y
where
V = volume of the footing
Pi,Py,p. = added mass in x-, y- and z-directions ^x> 9s = damping coefficient in x-, y- and
z-directions.
The hydrodynamic moments on the structure follow simply f r o m these forces. When spherical (or nearly spherical) components are considered the determina-tion of the hydrodynamic forces i^ similar lo that given f o r tubular segments (see section 2.3), in which the damping coefficient follows f r o m the drag of the body moving at a constant speed equal to the Root Mean Square of the relative velocity of the considered com-ponent of the structure.
4 Wave induced motions
4.1 Calculation of the motions
I n the preceding sections the forces exerted by the water on the floating structure were determined. The underwater part of the structure was divided into three-dimensional bodies (as for example the footings) and into tubular components, which were subdivided into small tube segments. Summation of the forces and resulting moments on these components yields the forces and moments acting on the structure in six degrees o f freedom. These forces and moments were determined in the Oxgyo^o set of axis (see fig. 5) having the Ox^yo plane in the free surface and the verdcal z-axis going through the centre of gravity G o f t h e structure. The forces and moments thus obtained are reduced to the forces (F^, Fy and F',) and moments {M^, My and M'.) acting i n the Gxyz set of axes.
Fig. 5. Definition of coordinate systems for tiie structure.
G is the centre of gravity
The sets of axes are fixed in space. The Oxoyo and Gxy planes are horizontal.
-12
The motions are obtained by solving the resulting six coupled linear simultaneous dilferendal equations
= Mx Z F ; = My E F ; = Mz I M ; = Ixx^ - I j I M ; = IyyÖ -lxy4> I M ; -l.x^ Where
M = mass of the floadng structure
= mass moment of inertia about the Gx-axis lyy = mass moment of inertia about the Gj-axis = mass moment of inertia about the Gz-axis I^y = = mass moment of inertia about the
Gx-and Gj-axes
4 i = = mass moment of inertia about the Gx-and Gz-axes
Jy. = hy = mass momcnt of inertia about the Gy-and Gz-axes
The resuldng equations of motion comprise a set of six coupled simuUaneous differential equations which may be expressed as follows
4 „ . . . . . . . y = i , 2 . . . 6 J=l where = X, Xl = >'. ^ 3 = z. ^ 4 = 4>, ^ 5 = e, Xe = 4'
F l , F2 and F3 represent the forces in the x-, y- and z-direcdons.
F 4 , F 5 and F g represent the roll, pitch and yaw
moments.
The wave elevations at 0 are defined as
C = -C„sinci>/
The resuldng motions are
X = —Xg sin (cor-fe,.) y = —y„ sin(üiï-|-Ej,) z = — Za sin(cor-l-ej)
<i> = -(^a sin (car-1-8^)
e = — sin ( t o t + Bg)
'/'
= -t/^o sin (cor-fe^)4.2 Some results obtained with the calculation procedure
From the calculation procedure given in the previous sections a computer programme was composed, which was used for some exploratory calculations on a floating drilling rig " N o r r i g " .
Some main particulars of this structure are given in fig. 6. On this platform rather extensive jtesting was performed, the results of which were published i n [6],
A part of the data presented is used f o r the validadon of the present calculation.
I n the calculations the footings of the floating struc-ture were replaced by spheres with the same volume as the actual footings. The added mass o f t h e spheres was assumed to be one half of the displaced mass. For the tubular members the added mass was taken equal to the displaced mass. These assumptions are i n principle only correct i n an unbounded fluid. Use being made of [4], the non-dimensional damping coefficient of the tubular members was estimated to be 1.17.
For the spheres an estimate was made,]use being made of [5]: Cc = 0.47.
Some dynamic characteristics of the p l a t f o r m are
/ , ^ = 2.58 10' k g f - m - s ' lyy =2.75 l O ' k g f - m - s ^ L, = 3.12 10' kgf-m-s^
I
These values led to the results given in the fig. 7 through 16. In fig. 7 the resulting mass coefficientspJi are com-pared with experimental data f r o m [6]. I n general the agreement is satisfactory. A n assessment of the validity of the damping coefficients q„,„ is difficult, since, due to the approach followed, a comparison between the calculated values and data f r o m captive (oscillation) tests is meaningless.
Some notes on these damping coefficients may be made, however. Fig. 8 shows the strong dependence of the dimensional damping coefficients on the wave height, i n particular when the frequencies| are low (long waves). Fig. 9 shows the significant influence o f the current.
L3
Fig. 6. Main particulars of the floating platform " N O R R I G " . A l l measures are given in metres
Translatory motions were measured at P Displacement = 24600 m^
14 EXPERIMENTS O WATERDEPTH 5 0 m a WMERDEPTH AO m -CALCULATED P11 P12 P13 0 " P21 P22 P23 ^ 0 e 0 P 3 I P32 P33 0 A-a— 0
1
1 *15 0 as 1.0 0 Q5 10 1.5 • F R E Q U E N C Y O F E N C O U N T E R I rad/s) -10 P15 P161
0 TO* J . P24 P25 P26 0 0 P 3 5 P 3 6 0 Q5 10 0 0 5 W 0 0 5 1.0 1.5 • F R E Q U E N C Y O F ENCOUNTER Irod/s) - 5 -10 i * 1 0 P4I hi P43 A fi JO & t 4 A 0 P 5 I P52 P53 è P61 P62 T63 0 0 0 5 10 0 0 5 10 0 0 5 10 15 • F R E Q U E N C Y O F E N C O U N T E R I rad/s) I 10 P u1
P45 Ï 5 •a P54 P5S 0 —è-èr -4-P65 P e s 0 @ O 0 5 10 O 0 5 10 O 0 5 10 1.5 • F R E Q U E N C Y O F E N C O U N T E R (rad/s)CALCULATIONS 5a= 1.00m
2.50 m
m V E DIRECTION 0 DEG
CALCULATIONS _
WITH CURRENT, p=0,v=1rn-s^ WAVE DIRECTION 0 D E G
• NO CURRENT Ea=1.60m
F R E Q U E N C Y OF ENCOUNTER imd/s} • F R E Q U E N C Y O F E N C O U N T E R I r a d / s l
17
WAVEDIREaiON O DEG
HEAVE EXCmNG FORCE AMPLITUDE PITCH EXCITING FORCE AMPLITUDE SURGE EXCITING FORCE AMPLITUDE
* ' FREQUENCY OF ENCOUNTER [ r a d / s l
Fig. 10. Calculated exciting forces and moments (wave direction 0 degrees).
0 WAVE DIRECTION 0 DEG
HEAVE PITCH SURGE
i 1 1 1 1 E q = 1.00 m, t,a = 1.80m Ea =2.50m
J
30 25 2.0 1,5 1.0 0.5 05 1.0 1.5 •Eq = 1.00 m E q = 1.80 m Ea = 2.50 m1
3.0 2.5 0 0.5 1.0 1.5 FREQUENCY OF ENCOUNTER f r a d / s i 2.0 a. i 1^ L U1
10 05 05 E q = 1.00 m E q = 1.80 m Eo = 2.50 m ;\
1.0 1,5WAVE DIRECTION 0 DEG
HEAVE PITCH SURGE
1
J
I
30 25 LU R 2,0 U J 1.0 0.5 NO CURRENT WITH CURRENT, P=0.V=1m.s' Ea= 1.80 mJ
3.0 r 2.5h 2.0 Q. 5 1.5 1.0 0.5 1 05 to 1.5 0 0.5 liO 1.5 FREQUENCY OF ENCOUNTER ( r a d / s j 0.5 1.0 1.5Fig. 12. Calculated effect of current on the motions (wave dhection 0 degrees, current direction 0 degrees).
WAVE DIRECTION 90 DEG
HEAVE ROLL SWAY
1 flOm 1 1 05 1.0 1.5 30 25 2.0: 1.5 1.0 o.5^ Ea = 1.80 m
1
]
0 0.5 1.0 1.5 FREQUENCY OF ENCOUNTER [ r a d / s ] 30 2.5 20 1.5 10 0.5 E a = J.80 m 0.5 1.0 1.519
WAVE DIRECTION 0 DEG
HORIZONTAL MOTION AT POINT P VERTICAL MOTION AT POINT P PITCH 30 2.5 2.0 1.5 1.0 0.5
^ 1
'Ea = 1.00 m £ • = i.eom Ea = 2.50 m\
\
( • ~ N 05 1.0 1,5 30 2-5 2.0 15 1.0 0.5 o [Ea = 1.00m ^ k a = i-eof + k a = 2.50m 0 0.5 1.0 1.5 FREQUENCY OF ENCOUNTER [ r a d / s l 1— G + ' E q = 1.00 m Ea = 1.80 m E q = 2.50 m — —u
0.5 10 1.5Fig. 14. Experimental and calculated motions (wave direction 0 degrees).
I n all cases the damping coefficients become very large around resonance frequency, when the motions are large. I n [6] it is shown among others that, contrary to the added mass and damping, the magnitude of the excidng force strongly depends on the water depth. The present calculadons are made f o r deep water. N o data obtained in sufficiently deep water are available for the validation of the present calculadon procedure, of which some results are presented in fig. 10.
From the evidence given in [6] it may be expected that the procedure followed here produces reasonable results. I n the fig. 11 through 14 the resulting motions of the platform are given. The fig. 11 and 12 show the effect of wave height and current on the motions. The significant effects on the damping coefficients results in minor changes of the motions. This shows the i n -significance of the damping on the motions o f the platform. Only in a very narrow frequency range around resonance non-linearities with the wave height occur. Fig. 13 shows that the motions do not signifi-candy change when the wave direction is changed to 90 degrees.
In fig. 14 a comparison is made between calculated motions and experimental data obtained on a scale model of the platform moored to chains in a model tank of the Netherlands Ship Model Basin [6].
The agreement appears to be satisfactory. I t has to be noted, however, that for a thorough validation of
the modon calculation procedure further investiga-tions are necessary. This holds i n particular f o r the calculated damping coefficients, which are important because of their strong relationship with the viscous part of the drift forces.
5 Viscous drift forces
5.1 Calculation of the drift forces
The drift force is the mean value of all the hydro-dynamic forces acting on the components. Since the damping force is the only force that contributes to the drift force in the present set-up, its magnitude is estiniated i n the following manner.
The components of the relative modon of a tube segment have been derived in section 2. The velocity components may be represented as follows
F^ = ~Vxi + Vxc cos cot + F „ sin a>t FJ, = — F,,, - I - cos U)t + SXTi CO ^
F . = - F , , - I - V.^ cos (Ot + F , j sin <ot
The damping force on the tube segment has the follow-ing magnitude
20
Then the drift force has approximately the following components acting i n the horizontal plane
= —qV^i cos^e + qVyi cos}'sin)'cos<5-l-+ qV,i cos a sin a cos
Dy = qVxi cos E sine cos K — ^F,,, C0s^7 + + cos a sin a sin ;S
The contribudon of the tube segment to the d r i f t moment about the vertical axis is
M, = X3Dy-y3Dx
The coefficient q is determined by
and
V' = V^, + K/i + K/i + iF,^, + i Kf, + i F / , +
In section 2.3 it has been shown that F , i = F „ c o s | i + F , c o s ^
FJ,, = F„,sin/i + F,sini8 = 0
The contribudons to the drift force and moment of the footings and other three-dimensional components are determined in a similar way.
5.2. Some resuhs obtained with the calculation procedure
The drift force due to waves, only is proportional to the mass transport velocity. This velocity is propor-tional to the wave height squared. The damping coefficients q also appear to increase with the wave height. These are the causes f o r the very rapid increase of the drift force with the wave height shown in fig. 15. In this figure some experimentally obtained values are also given. I n general the measured and calculated values are of tbe same order of magnitude. A t some higher frequencies the agreement may even be
consid-ered reasonable. ' It is to be noted, however, that the drift forces, given
in the graph, were estimated f r o m anchor chain forces measured during model tests.
The measurement of these forces is in general rather " iTiö:curate. The conversion of the anchor chain forces to drift forces is not accurate either. Further invesdga-tions are therefore needed for a better validation of the subject calculation methods.
Finally fig. 16 shows the influence of current on the drift force. In the diagram on the left the total drift force is given as experienced by the structure due to waves only and that due to waves and current. I n the diagram on the right the two curves represent the drift force due to waves. However, the solid curve was cal-culated with damping coefficients which are obtained when both waves and current are considered. This diagram shows again the importance o f the current velocity on the damping coefficient and thus on the
drift force. , 25000 e t Q 2 20000, 15000 10000 SOOOi X-FORCE WAVE DIRECTION 0 DEG
0 ^ 0 = 1.00 m
^ X E q = 1.80 m , r =250m
1.0 1.5 FREQ OF ENCOUNTER [rad/s]
25000 UJ 20000: I - i S 15000 • 10000 5000 Y-FORCE WAVE DIRECTION 90 DEG
A
f
y
'0 0.5 1.0 15 — » - FREQ OF ENCOUNTER [rad/s) Fig. 15. Effect of wave height and wave direction on the drift force. Comparison with experimental data.
250000
X-FORCE WAVE DIRECTION 0 DEG
Ea = 1.eOm
. WAVES ANO CURRENT T|=0, V= 1 m/s NO CURRENT WAVES ONLY 25000 I 150:;: z
a
A 10000 5000X FORCE DUE TO WAVES WAVE DIRECTION 0 DEG
Ea = 1-80m • WAVES AND CURRENT
'q = O . V = T m / s NO CURRENT WAVES ONLY, S E E L E F T SIDE
/ /
1^ / / / / 0.5 1.0 1.5 FREaOF ENCOUNTER [ r a d / s lFig. 16. Calculated effect of current on the drift force.
6 Concluding remarks
The motion calculation procedure presented i n the present report differs f r o m those developed by others [2] and [3] in the calculation of the damping coefficient. The present approach enables the determination of the viscous part of the drift forces, caused by the mass transport in waves. The results show that the drift force thus determined is of the same order of magni-tude as the values obtained experimentally. This leads to the supposition that this viscous drift force cannot be neglected.
References
1. H s u , F . H . and K . A . BLENKARN, Analysis of peak mooring force caused by slow vessel drift oscillation in random seas. Proceedings of the Offshore Technology Conference, 1970, Dallas (Texas), U S A .
2. PAULLTNG, J . R . , Wave induced forces and motions of tubular structures. Paper presented at the 8th O N R Symposium on Naval Hydrodynamics, Pasadena, U S A . August 1970. 3. HooFT, 3. P., Hydrodynamic aspects of semi-submersible
platforms. Thesis, Delft. The Netherlands, 1972.
4. PAAPE, A . , H . N . C . BREUSERS and J . D . VAN DEN BUNT, L'estimation des forces hydrodynamiques sur les pieux, published in: Connaissance de la houle, du vent, du courant pour le calcul des ouvrages pétroliers. Publication de iTnstitut Francais du Pétrole, no. 16, Paris, France, 1970.
5. HOERNER, S., Fluid Dynamic Drag. Second edition, 1965. 6. HOOFT, J . P., The behaviour of a five column floating drilling
unit in waves. Netherlands Ship Research Centre T N O , Report nr.: 156 S, Delft, The Netherlands, November 1971.
Appendix I 0, X
In section 2.2 it is assumed that the force on the tube due to a velocity x along the x-axis is (see fig. A l )
Fx = qx cos £
z
Fig. A l . Forces caused by surge velocity.
The components of this force in the x-, y- and z-direc-tions are
- -<7XC0S^£
= qx COS £ sin s cos K
F- = qx COS e sin £ sin K
Likewise the components of the force caused by a velocity y along the ^-axis are (see fig. A2)
F^= ^j? cosy sin y cos <5
F,, = - ^ y cos^y
F, = ^ j p cosy sin V sin 5
Y
Z
Fig. A 2 . Forces caused by sway velocity.
The components of the force caused by d velocity z along the z-axis are (see fig. A3)
Fx= ^z cos a sin a cos ^ Fy = qz cos a sin a sin fi F^ = —qzcos^a
1
zFig. A 3 . Forces caused by heave velocity.
The added mass forces caused by accelerations in the
Appendix I I
The total force exerted by the water on a segment of a submerged tube may be summarized as follows
= i
a , j ( p X j + qxj) + i:,e-''-'-Wi f o r i = 1,2,3, ...,6Where x , = x, = ƒ , JCa = z, = <j), X^-O, X ^ - ^ I / and Z j = depth of the segment below the water surface.
K„ K2 and K2 represent the forces in the x-, y- and z-directions respecdvely while K4., and represent the roll, pitch and yaw moments. The coefficients and W, are given by the following matrices and relationships.
For / = 1, 2, 3 and J= 1, 6
-cos^e cos 7 sin y cos (5 cosasinacos^ j'3 cos a sin a cos/? —X3 cos a sin a cos ^ . X j c o s y s i n y c o s ó
•Z3 cosy siny COS)? —Z3 cos e +y3 cos''e
c o s e s m e c o s / c — c ö s y cosasinasin^ ;'3Cosasinasin — J C 3 C o s a s i n a s i n i S —x^cos^y + Z3 cos 7 -!-Z3 COSE s i n s COSK — ƒ 3 c o s e s i n e c o s K c o s E s i n E s i n K c o s y s i n y s i n ó - c o s ^ o ; —>'3Cos a X3COS a+ A:3 cos 7 sm y sm/c — Z3 cosy sin 7 sin 5 Z3 cose sine sin JC —y3 COSE sine sin K
For ( = 4, 5, 6 a n d j = l , 6
W\ = Q9 •M•k•cos|lCxy+a^^{-pco^ Cos / i C^,+q(o cos p S^y) + a 1 zC - P(o^ sin p C^y+qo)s\np S ) -I- (J, 3 ( - pw^S^, -I- g w C „ )
TlD
= 0 3 — 4 — h l - k - % \ n p C x y + a2y{-pwrcospCxy + q(acospSxy) + a22{-p<o s\npCxy+q(os\npSxy)-\-a23{-p(o Sxy + qoiCxy)
TtD^
^3 = Q9 - 4 - •M-Sxy + ii3i(-po}^cospCxy + qcocospSxy) + a32(-'P(o^&int^Cxy + 9(osmp S^y) + a 3 3( - pco^S^y + qcoC^y)
Where
W4=y3W3-Z3W2
W,=Z3W,-X3W3
W6=X3W2-y3W,
C^y = cos(A-X3 cos/t-l-A-j'jsin/i-cüO Sxy — sin (^^-3 cosp+ky3 sin p-cot)
The hydroslatic force's which result f r o m a Cube piercing the water surface at J : , , r j , ; 0; are Where ^ / - T
C,.,^v+C„M^.'
f o r 1 = 1,2,3,4,5,6 J 3 l 89-0, 0 0' 0 0 '0 10 0 0 sin o{ ' - V , sin a 0 0 sin %sin Ot 16 \ sin a sin'a / c o s V ^ sinV? •sin a sin a Ö 0 0 0 4sina sin^P
^3 —— I y I sly + — cos'jS + - f /< sin/1 C.i, 4srna V 16 \ Sih a Q9 4 sin a P' cos^fi 16 \ stn a /ccos/(C',
i \
7
cos (kxi cos / ( + /cy 1 sin /< - cur)sin/;/cxj c o s / i - I - ( , sin/<'i=ï w/) sin, a sin iz /sin^jS cos^/J sin a 16 '0
+
sin a sin^'a 0 ,0 0 0PUBLICATKW iS OF T H E N E T H E R L A N D S SHIP RESEARCH C E N T R E T N O L I S T O F E A R L I E R P U B L I C A T I O N S A V A I L A B L E O N R E Q U E S T
P R I C E P E R C O P Y D F L . lOi— ( P O S T A G E N O T I N C L U D E D )
M = engineering department S =• shipbuilding department C = corrosion and antifouling departmettt
Reports
90 S Computation of [pitch and heave motionsTor arbitrary ship forms. W. E . Smith, 1967.
'911 M Corrosion in exhaust driven turbochargers on marine diesel engines using heavy fuels. R . W. Stuart Mitchell, A. J . M . S. van Montfoort and V . A . Ogale, 1967.
95 M Residual fuel treatment on board ship. Part I I . Comparative cylinder wear measurements on a laboratory diesel engine using filtered or centrifuged residual fuel. A de Mooy, M . Verwoest and G . G . van der Meulen, 1967.
'93 C Cost relations of the treatments of ship hulls and the fueT. con' sumption of ships. H . .1. Lageveen-van Kuijk, 1967.
94 C Optimum conditions for blast cleaning of steel plate. 1 Rem-melts, 1967.
•95 M Residual fuel treatment on board ship. Part I . The effect of cen-trifuging, filtering and homogenizing on the unsolubles in residual fuel. M . Verwoest and F . J . Colon, 1967.
96 ïS Analysis of the modified strip theory for the calculation of ship. motions and wave bending moments. J . Gerritsma and W. Beu-kelman, 1967.
97 S O n the efficacy of two different roll-damping tanks. J . Bootsma and J . J . van den Bosch, 1967.
98 S Equation of motion coefficients for a pitching and heaving des-troyer model. W. E . Smith, 1967.
99 S The manoeuvrability of ships on a straight course. J . P. Hooft, 1967.
100 S Amidships forces and moments on a C ^ = 0.80 "Series 60'' model in waves from various directions. R . Wahah, 1967. 101 C Optimum conditions for blast cleaning of steel plate. Conclusion.
J . Remmelts, 1967.
102 M The axial stiffness of marme diesel engine crankshafts. Part I . Comparison between the results of full scale measurements and those of calculations according to published formulae. N . J . . Visser, 1967.
103 M The axial stilTness of marine diesel engine crankshafts. Part I L . Theory and results of scale model measurements and comparison with published formulae. C . A . M . van der Linden, 1967. 104 M Marine diesel engine exhaust noise. Part I . A mathematical model.
J . H . Janssen, 1967.
105 M Marine diesel engine exhaust noise, Part I I . Scale m o ü e i s of exhaust systems. J . Buiten and J . H . Janssen, 1968.
106 M Marine diesel engine e,\haust noise. Part I I I . Exhaust sound criteria for bridge wings. J . H . Janssen en J . Buiten, 1967. 107 S Ship vibration analysis by finite element technique. Part I .
General! review and application to simple structures, statically loaded. S. Hylarides, 1967.
rOS'M Marine refrigeration engineering. Part I . Testing of a decentral-ised refrigerating installation. J. A . Knobbout and R . W. J . Kouffeld, 1967.
109 S A comparative study on four different passive roll damping tanks. P a r t i . J . H . Vugts, 1968.
A I O S Strain, stress and flexure of two corrugated and one plane bulk-head subjected to a lateral, distributed load. H . E . Jaeger and P. A . van Katwijk, 1968.
111 M Experimental evaluation of heat transfer in a dry-cargo ships' tank, using thermal oil as a heat transfer medium. D . J . van der Heeden, 1968.
112 S The hydrodynamic coefficients for swaying, heaving and rolhng cylmders in a free surface. J . H . Vugts, 1968.
113 M Marine refrigeration engineering. Part I I . Some results of testing a decentralised marine refrigerating unit with R 502. J . A . K n o b -bout and C . B . Colenbrander, 1968.
114 S The steering of a ship' during the stopping manoeuvre., J , P. Hooft, 1969.
115 S Cylinder motions in ibeam^ waves. J . H , Vugts, 1968.
116 M Torsional-axial vibrations of a ship's propulsion system. Part I . Comparative investigation of calculated and measured lorsional-axial vibrations in the shafting of a dry cargo motorship. C . A . M . van der Linden, H . H . 't Hart and E . R . Dolfin, 1968. 117 S A comparative study on four different passive roll damping
tanks. Part U . J . H . Vugts, 1969.
118 M Stem gear arrangement and electric power generation in ships propelled by controllable pitch propellers. C . Kapsenberg, 1968.
119 M Marine diesel engine exhaust noise. Part IV'. Transferdamping data of 40 modelvariants of a compound resonator silencer.. J . Buiten, M . J . A . M . de Regt and W . P. Hanen, 1968. 120 C Durability tests with prefabrication primers in use steet of plates.
A., M . van Londen and W. Mulder, 1970.
12I.S Proposal for the testing of weld metal from the viewpoint o f brittle fracture initiation. W. P. van den Blink and J . J . 'VV. Nib-bering, 1968.
122 M The corrosion behaviour of cunifer 10 alloys in seawalerpiping-systems on board ship. Part I . 'W. J . J . Goetzee and F . J . Kievits, 1968.
123 M Marine refrigeration engineering. Part I I I . Proposal for a specifï-cation of a marine refrigerating unit and test procedures. J . A . Knobbout and R. 'VV. J . Kouffeld, 1968.
124 S The design of U-tanks for roll damping of ships. J . D . van ,den Bunt, 1969.
125 S A proposal on noise criteria for sea-going ships. J . Buiten, 1969'. 126 S A proposal for standardized measurements and annoyance rating of simultaneous noise and vibration in ships. J . H . Janssen, 1969. 127 S The braking of large vessels I I . H . E . Jaeger in collaboration with
M . Jourdain, 1969.
128 M Guide for the calculation of heating capacity and healing coils for double bottom fuel oil tanks in dry cargoi ships. D . J . van der Heeden, 1969.
129 M Residual fuel treatment on board ship. Part I I L A . de Mooy, P. J . Brandenburg and G . G . van der Meulen, 19691
J30 M Marine diesel engine exhaust noise. Part V . Investigation of a double resonatorsilencer. J . Buiten, 1969.
131 S Model and full scale motions of a twin-hull vessel. M . F . van Sluijs, 1969.
132 M Torsional-axial vibrations of a ship's propulsion system. Part I I . W. van Gent and S. Hylarides, 1969.
133 S A model study on the noise reduction etfect of darhpirig layers aboard ships. F . H . van T o l , 1970.
134 M The corrosion behaviour of cunifer-10 alloys in seawaterpiping-systeras on hoard ship. Part I I . P. J . Berg and R . G . de Lange. 11969.
135 S Boundary layer control on a ship's rudder. J . H . G . Verhagen, 1970.
136S Observations on waves and ship's behaviour made on board of Dutch ships. M . F . van Sluijs and J . J . Stijnman, 1971. 137 M Torsional-axial vibrations o f a ship's propulsion system. Part I I I .
C . A . M . van der Linden, 1969.
138 S The manoeuvrability of ships at low speed.. P. Hooft and M. W. C . Oosterveld, 1970.
139 S Prevention of noise and vibration annoyance aboard a sea-going passenger and carferry equipped with diesel engines. Part I . Line of thoughts and predictions. J . Buiten, J . H . Janssen, H . F . Steenhoek and L . A . S. Hageman, 1971.
140 S Prevention of noise and vibration annoyance aboard a sea-goirig passenger and carferry equipped with diesel engines. Part I I . Measures applied and comparison of computed values with measurements. J . Buiten, 1971.
141 S Resistance and propulsion of a high-speed single-screw cargo liner design. J . J . Muntjewerf, 1970.
142 S Optimal meteorological ship routeing. C . de Wit, 1970'., 143 S Hull vibrations, of the cargo-liner "Koudekerk". H . H . 't Hart,.
1970.
144 51 Critical consideration of present huU vibrationianalysiS'. S. H y l a -rides, 1970.
145 S Computation of the hydrodynamic coefficients of oscillating cylinders. B . de Jong, 1973.
146 M Marine refrigeration engineering. Part I V . A Comparative stuyd on single and two stage compression. A . H . van der T a k , 1970i 147 M Fire detection in machinery spaces. P. J . Brandenburg, 1971. 148 S A reduced method for the calculation of the shear stiffness o f a
ship hull. W. van Horssen, 1971.
149 M Maritime transportation of containerized cargo. Part I I . Experi-mental investigation concerning the carriage of green coffee from Colombia to Europe in sealed containers. J . A . Knobbout, 1971, 15.Ü'S The hydrodynamic forces and ship motions in oblique waves-.
151 M Maiitime transportation of containerized cargo. Part I .
Theoretical and experimental evaluation of the condensation risk when transporting containers loaded with tins in cardboard boxes. J . A . Knobbout, 1971.
152 S Acoustical investigations of asphaltic floating floors applied on a steel deck. 3. Buiten, 1971.
153 S Ship Y'bration analysis by finite element technique. Part I I . Vibra-tion 'inalygllp^j^kgdes, 1971.
155 M M:mne d i e j e j g ^ ^ e x h a u s t noise. Part V I . Model experiments on the influeijee^Pihe shape of funnel and superstructure on the radiated exhaust soun(|. J . Buiten and M . J . A . M . de Regt, 1971. 156 S The bchavioiitagf a five-column floating drilling unit in waves.
J . P. Hooft, 1971.
157 S Computer prostamsKfor the design and analysis of general cargo ships. J . Holtcop, 1971.
158 S Prediction o S s h i p manoeuvrability. G . van Leeuwen and J . M . J . . l o u r ^ i S f ? ^
159 S D A S H compu^r program for Dynamic Analysis of Ship Hulls. S. H y l a r i d e s j ^ l .
160 M Marine ref|)Mpjjaa^engineering. Part V I I . Predicting the con-trol
propertielSifi^Ka
valves in marine refrigerating installations A . H . van der T a k , 1971.161 S Fuil-^cale measiu-ements of stresses in the bulkcarrier m.v. 'Ossandrecht'. 1st Progress Report: General introduction and information. Verification of the gaussian law for stress-response to waves. F . X . P. Soejadi, 1971.
162 S Motions and mooring forces of twin-hulled ship configurations. M . F . van Sluijs, 1971.
163 S Pcrtjnnance and propeller load fluctuations of a ship in waves. M . ^ van Sluijs, 1972.
164 S T h e « f B c i e n c y of rope sheaves. F . L . Noordegraaf and C . Spaans, 1972.
165 S Stress-analysis of a plane bulkhead subjected to a lateral load. P. M^ijers, 1972.
166 M CoBftiarotating propeller propulsion. Part I , Stern gear, line shaft system and engine room arrangement for driving contra-rotating propellers. A . de Vos, 1972.
167 M Contrarotatmg propeller propulsion. Part I I . Theory of the dynajBiic behaviour of a line shaft system for driving contra-rotajjng propellers. A . W . van Beek, 1972.
169 S A n a l ^ i s of the resistance increase in waves of a fast cargo ship. J . Gerritsma and W . Beukelraan, 1972.
170S Simulation of the steering- and manoeuvring characteristics of a second generation container ship. G . M . A . Bnmimer, C . B . van d_e Voorde, W . R . van Wijk and C . C . Glansdorp, 1972. 172 M R e l i a ï i l i t y analysis of piston rings of slow speed two-stroke
marine diesel engines from field data. P. J . Brandenburg, 1972. 173 S Wave load measurements on a model of a large container ship.
Tanr^engGie, 1972.
174 M Guide for the calculation of heating capacity and heating coils for deep tanks. D . J . van der Heeden and A . D . Koppenol, 1972. 175 S Some aspects of ship motions in irregular beam and following
waves. B . de Jong. 1973.
176 S Bovv^lare induced springing. F . F . van Gunsteren, 1973. 177 M Mari;gjne transportation of containerized cargo. Part I I I . Fire
tests m closed containers. H . J . Souer, 1973. 178 S Fracture mechanics and fracture control for ships.
J . J . W. Nibbering, 1973.
179 S Effect of forward draught variation on performance of full ships. M . F . van Sluijs and C . Flokstra, 1973.
182 S Finite element analysis of a third generation containership. A. W. van Beek, 1973.
184 S Numerical and experimental vibration analysis of a deckhouse. P. Metiers, W. ten Gate, L . J . Wevers and J . H . Vink, 1973. 185 S F u l l | c a l e measurements and predicted seakeeping performance
of the containership "Atlantic Crown". W . Beulcelman and M. Buitenhek, 1973.
186 S Wavgs induced motions and drift forces on a floating structure. R . Wahab, 1973.
Communications
15 M Refrigerated containerized transport (Dutch). J . A . Knobbout, 1967.
16 S Measures to prevent sound and vibration annoyance aboard a seagoing passenger and carferry, fitted out with dieselengines (Dutch). J . Buiten, J . H . Janssen, H . F . Steenhoek and L . A . S. Hageman, 1968.
17 S Guide for the specification, testing and inspection of glass reinforced polyester structures in shipbuilding (Dutch). G . Hamm, 1968.
!8 S A n experimental simulator for the manoeuvring of surface ships. J . B . van den Brug and W . A . Wagenaar, 1969.
19 S The computer programmes system and the N A L S language for numerical control for shipbuilding. H . le Grand, 1969.
20 S A case study on networkplanning in shipbuilding (Dutch). J . S. Folkers, H . J . de Ruiter. A . W . Ruys, 1970.
21 S The effect of a contracted time-scale on the learning ability for manoeuvring of large ships (Dutch). C . L . Truijens, W . A . Wage¬ naar, W . R . van Wijk, 1970.
22 M A n improved stern gear arrangement. C . Kapsenberg, 1970. 23 M Marine refrigeration engineering. Part V (Dutch). A . H . van der
T a k , 1970.
24 M Marine refrigeration engineering. Part V I (Dutch). P. J . G . Goris and A . H . van der T a k , 1970.
25 S A second case study o n the application of networks for pro-ductionplanning in shipbuilding (Dutch). H . J . de Ruiter, H . Aartsen, W . G . Stapper and W . F . V . Vrisou van E c k , 1971. 26 S O n opthnum propellers with a duct of finite length. Part I I .
C . A . Slijper and J . A . Sparenberg, 1971.
27 S Finite element and experunental stress analysis of models of shipdecks, provided with large opetwngs (Dutch). A . W . van Beek and J . Stapel, 1972,
28 S Auxiliary equipment as a compensation for the eff'ect of course instability on the peiformance of helmsmen. W . A . Wagenaar, P, J . Payraans, G . M . A . Brummer, W . R . van Wijk and C . C . Glansdorp, 1972.
29 S The equilibrium drift and rudder angles of a hopper dredger with a single suction pipe. C . B . van de Voorde, 1972.
30 S A third case study on the application of networks for production-planning in shipbuilding (Dutch). H . J . de Ruiter and C . F . Heij-nen, 1973.
31 S Some experiments on one-side welding with various backing materials. Part I . Manual metal arc welding with coated electro-des and semi-automatic gas shielded arc welding (Dutch). J . M . Vink, 1973.
32 S The application of computers aboard ships. Review of the state of the art and possible future developments (Dutch). G . J . Hoge-wind and R . Wahab, 1973.
33 S F R O D O , a computerprogram for resource allocation in network-planning (Dutch). H , E , I . Bodewes, 1973.
34 S Bridge design on dutch merchant vessels; an ergonomie study. Part I : A summary of ergonomie pomts of view (Dutch). A . Lazet, H . Schufi'el, J , Moraat, H , J . Leebeek and H . van D a m ,
1973.
35 S Bridge design on dutch merchant vessels; an ergonomie study. Part I I : First results of a questionnaire completed by captains, navigating officers and pilots. J . Moraal, H . Schuffel and A , Lazet, 1973.
36 S Bridge design on dutch merchant vessels; a n ergonomie study. Part I I I : Observations and preliminary recommendations. A . Lazet, H . Schuff'el, J . Moraal, H . i. Leebeek and H . van D a m , 1973.
37 S Application of finite element method for the detailed analysis of hatch corner stresses (Dutch), J . H , Vink, 1973.
