DYNAMIC BEHAVIOUR OF FAST
MONOHULLS
J.A. Keuning
F.H.H.A. Quadvlieg Report No. 977-P
International Conference on High Speed Passenger
Craft-Future
Developments and the NordicInitiative, RINA, 15-16 June 1993, London
Deift University of Technology Ship Hydromechanics Laboratory Mekelweg 2
2628 CD DeIft The Netherlands
INTERNATIONAL CONFERENCE
on
HIGH SPEED PASSENGER
CRAFT
-FUTURE DEVELOPMENTS
AND THE
NORDIC INITIATIVE
15 16 JUNE 1993 LONDON
PAPERS
© 1993 The Royal Institution of Naval Architects
The Institution is not, as a body, responsible for the opinions expressed by the individual authors or speakers.
THE ROYAL INSTITUTION OF NAVAL ARCHITECTS 10 Upper Beigrave Street,
London, SW1X 8BQ
RINA
INTERNATIONAL CONFERENCE on
HIGH SPEED PASSENGER CRAFT
-FUTURE DEVELOPMENTS AND THE NORDIC INITIATIVE
LONDON
)
SUMMARY
A non-linear computational model has been developed to calculate the heave and pitch motions of planing ships in irregular head waves.
The computational model is based
on the previous model as presented by
Zarnick [1] in 1979. This model has been extended significantly to be able to
handle the behaviour of planing craft in irregular waves. For this purpose a
"wavemaking" routine has been added which makes it possible to calculate the
wave elevation along the length of the ship any time corresponding
to a
predetermined wave spectrum.
In addition polynomial expressions for sinkage and trim of the craft at speed
have been developed based on an extensive series of model experiments with a
systematic series of planing hull forms. In a new computational routine the
resulting sinkage and trim are being used to asses the pressure distribution over the length of the planing model at speed. This determines the relative magnitu-de of the hydrostatic and hydro dynamic part of the lift.
The formulations used for the determination of the added mass and the wave
exciting forces have been examined and extended to become dependent to the
geometry of the craft in actual contact with the waves while performing large
relative motions.
A computer code has been developed to solve the non-linear equations in the time domain. The results have been extensively verified with experimental
results derived from tests in the Deift Towing Tank.
Finally it will be shown that a proper formulation of the limiting criteria with
respect to workability of a craft is of great importance on the final outcome of
an optimisation and this determines whether a linear or a non-linear calculation method may be used.
1
- INTRODUCTION
During the last decades a growing interest in fast ships may be noticed. In par-ticular the possible fulfilment of existing needs in the field of fast transport of light and expensive cargo, the transportation of passengers at high speeds and
the possible use for surveillance and patrol functions has drawn considerable
interest of both shipowner, designers and researchers to faster ships. All kind
of so called "advanced" concepts have been developed of which the planing craft, the catamaran, the air cushion vehicle, the surface effect ship, the small
waterplane area twin hull ship and the hydrofoil should be mentioned.
All these concepts were aimed at high speeds and possibly
high sustainedspeeds in a seaway. Compared to the planing boat all other concepts are more complicated and often in the need for a "ride control" system to improve their
seakeeping behaviour. This tends to make these concepts more complicated and
therefore usually more expensive. In particular the reliability of the ship seen from an operating point of view, makes these more complicated concepts
economically less attractive. Therefore there remains a place for the relative "simple" and well-proven concept of the planing boat, i.e., the fast monohull,
in the foreseeable future. Considerable effort however should be put in optimi-sing the seakeeping capabilities of the fast planing monohulls.
From experience it is known that the heave and pitch motions and more in
par-ticular the vertical (bow) accelerations aboard the ships in head seas are the limiting factors with respect to the seakeeping capabilities of planing boats.
Therefore the need for a suitable prediction tool for these motions and
acce-lerations of these fast monohulls in irregular waves is obvious to be able to
optimise their seakeeping behaviour and with that their operability.
This prediction tool should properly incorporate the effect of the prime para-meters and phenomena so particular for the motions in waves of these planing boats. These were considered to be, amongst others:
- the deadrise distribution over the length of the hull.
- the bow shape and flare, more particular the geometry of the hull above the still waterline.
- the dynamic lift developed by the planing bottom. - the sinkage and trim of the craft at speed.
- forces related to the large relative motions. - the effects of high forward speeds.
Most of these parameters and phenomena
are the cause of important
This makes the use of the commonly used
calculation routines for shipsmoving with forward speed in waves, which are based on the so called linear strip theory method, less applicable.
So a computational model has been developing which takes those effects into
account. It was in the scope of this project to derive for all
phenomenainvolved an exact formulation for use within the foreseen setup of the present
computational model, but emphasis has been placed on a qualitative correct prediction of the effect of the predominant parameters under consideration on
the performance of the planing hull in waves.
2- COMPUTATIONAL MODEL
The basis of the presented non-linear computational model is formed by the work of Zarnick [1] as published in 1979. He formulated the computational
model for the heave and pitch motions of a hard chine planing craft with
constant deadrise over the length of the ship in regular head waves.
A short summary of this model will be presented here. For a more detailed
description of the computational model reference is made to [2].
The model is a non-linear strip theory approach, in which the hull is divided
over the length in a number (i.e., 20) of transverse strips with assumed
constant cross sectional shape. The coordinate used is presented in Figure 1,representing the ship in a steady state equilibrium position at constant forward
speed.
Figure 1 Coordinate system used
Although the mathematical justification for using a modified striptheory is not very rigorous, both Zarnick and Martin [3] obtained results for the motions in
regular waves which compared well with measured values.
in which: T = represents the towing force,
D = the drag force
N = the hydrodynamic lilting force BL = the buoyancy force
M = the mass of the ship
McG
= Tcos(O+) - Nsin6 - DcosOMcG =
-Tsin(O+) - NcosO - BL + DsinO + W (1)ii
= Tx +Nx +BLXb -Dxd
By assuming constant forward speed, the acceleration along the X-axis may considered to be zero. In the present formulation this simplification will be
made, although its omittance is a development foreseen in the near future. The total lift force on a strip of the ship is determined by:
dF = (-j(maV)
CD,c pbV2 )cosO _abfpgA (2)in which: dF = the force per strip
the added mass of the strip
v = the vertical velocity in the plane of the cross section
= cross flow drag coefficient
b = instantaneous half beam of the strip, including "pile up"
abf = buoyancy correction coefficient
A = instantaneous submerged cross sectional area
g = acceleration due to gravity
The buoyancy correction coefficient accounts for the difference between the actual pressure distribution over the submerged part of the hull, composed of hydrodynamic and hydrostatic pressure, when compared with the "lift" force
arising from an assumed hydrostatic pressure distribution over the submerged
volume.
The velocity components U and V of the relative motion of the water along the
hull resulting from the forward velocity of the craft, the heave and pitch
motions and the orbital velocity in the waves may be written as:
total vertical force total pitching moment total added mass actual pitch angle
vertical component orbital velocity length coordinate body fixed axis added mass correction factor
in which: -Ncoss-BL = NxC+BLxb = Ma = 0 = = ka = U = cos0 - (cG-wZ)sinO (4) where:
wz the orbital velocity of the water, zi = the instantaneous position of strip i,
= the position of the strip.
According to Wagner's formulation the dynamic lift is considered to originate
from the change of momentum of the incoming fluid and a cross flow drag
force on the section.
Elaboration of the equations for the dynamic and static lift forces and moments
yields:
-Ncos0 -BL = -Msin0cos0 -Mcos20 +QaOcosO
dw
+ cosO [-Ma0(cos0-zs1n0)+jma dc05O -jmawz0sm0dE
3m 3m
_Jv
dE-jUma0dE +Juvad
JcD,CPbVd ] _abfPgJAd Nx +BLxb = Qa .sin0+ Qa cos0 'a 0
dw- 1 Qa
0(cos0 -sm0) + J ma zcos0Ed-
J maw 0s1n0dE 3m . 3m-
Jv
Ed-
J Uma0dE + J UV aIn the formulations the instantaneous wetted beam and submerged cross
sec-tional areas of the sections are being used resulting from the actual
momenta-neous relative motion of each section as a combined result of heave, pitch and
wave elevation. This implies that the wave elevation at each time along the length of the forward moving craft must be known in irregular waves. Hereto
a wave generator is being build into the program, using either the well-known
energy distribution in the frequency domain according to the formulations of
Pierson-Moskowitz or a measured spectrum. The irregular wave train is being simulated using a random phase model according to:
N
(x,t) =
a1 sin(wt+k+)
(7)i=1
In addition also the vertical and horizontal orbital velocities are being
calcula-ted as a function of time (t) and place (x) along the length of the model using
N coshk(h+z) u(x,t) =
sin(wt+k+4)
i=1 sinhkh N sinhk,(h +z) v(x,t) = wacos(wt+k+4)
i1
sinhk1hin which: N = number of components
aj
=
amplitude of th component= frequency of th component
=
wave number ith componenth = water depth
phase
Using the well-known dispersion relation for deep water
waves, the time
history of the wave elevation is generated using a random phase generator forthe phase of the different components.
In the simulation program the vessel is moving with a constant speed against
these waves.
2.1
Steady State Planing Condition
In the case of ships advancing at relative high forward speeds, it is known that
the sinkage and trim of the craft may become quite substantial. The radical change in reference position,
i.e., the position around which the craft is
supposed to perform its wave induced motions, implies among other things the following modifications with respect to the equations of motion describing the dynamic behaviour of the craft in waves:
- the existence of dynamic lift resulting from the high relative
water velocity over the inclined bottom
- a change of the submerged geometry of the craft when compared to the zero trim, zero sinkage and zero speed situation.
Both effects account for part of the non-linear behaviour of the craft at speed
in waves and are not accounted for in the usually used linear theory. The intro-duction of these non-linearities into the computational model may be performed in different ways:
First, one could account only for the change in reference position (sinkage and trim) and use the changed geometry as an input in the otherwise linear
calcula-tions. This has been done among others by Beukelman [41 and it yields im-proved correlation with model experiments in comparison with calculations using the zero speed reference position. It is obvious that the introduction of
the reference position in this manner only accounts for the change in geometry
of the submerged hull and not for the change in
pressure distribution. SeeFigure 2. 20 kn
Pitch
20 10d/
0/
/
/
/
/
\Tr
4.48°/
witi reevenc posRov ct 54eec Trii-n -o.V' with zero s,eedreerec position
Figure 2. Influence of reference position on the pitch motion of planing
craft, using linear strip theory.
The reference position of the craft at speed is obtained in this particular case
from results of a model experiment.
Another method was followed by Zarnick. He made use of the empirical
for-mulations for the pressure distribution over the length of theplaning bottom as
derived from model experiments with planing wedges. These expressions have
been formulated among others by Sottorf [5], Sedov [6], Shufold [7]. Savitsky [8] used these formulations for the determination of the running
trim of a
planing craft at speed. Zarnick derived the unknown coefficients al0f, ka andCD,C using these formulations and used those values in the equations of motion
describing the calm water condition, i.e., by omitting any wave induced term.
By doing so a steady state equilibrium position is obtained, which is used as
the reference position for the motions of the craft in waves. The validity of the
values of the coefficients obtained from planing wedges with mostly constant
deadrise remains uncertain and the results of sinkage and trim obtained by this method are often in no good agreement with measured values.
In the present computational model the procedure is reversed: instead of
deter-mining abf, CD,C and ka from experiments with planing wedges, the value of these coefficients is determined using a calculated value for the sinkage and trim of the particular planing craft under consideration at a given speed. The sinkage and trim are obtained from polynomial expressions derived from an
extensive series of model tests with a large systematic series of planing hull forms. The parent model of this series was the 12.5 degrees deadrise model
used by Clement and Blount [9] who tested this model in 1963 with 5 different
length to beam ratio's, 4 different displacements and 4 different longitudinal positions of the centre of gravity. This systematic series has been extended by Keuning and Gerritsma in 1982 with a similar series with 25 degrees deadrise
[10] and in 1992 with 30 degrees deadrise [11].
The bodyplans of the three parent models of these series
are presented inFigure 3.
12.5 25.0 30.0
Figure 3 Bodyplans of the 12.5, 25.0 and 30.0 degrees deadrise
The parameters used in the polynomial expressions are:
- the length to beam ratio over the chine
- the weight of displacement related to the projected area of the planing
bottom
- the deadrise angle at midship
- the longitudinal position of the centre of gravity.
The polynomial expressions have been derived using
a least square fitting
routine through the data for a number of specific volumetric Froude numbersand have the following fonn:
L
L2
L3
AO(Fflv) =a0+a1+a2() +a3(_) +a4
V2'3
6(,3)
RCG (Fnv) V1 /3 AAL
( )_.+a .,±LCG+a7LCG+a8LCG2 +a9LCG3 +a10(
VV3 v2'3 B '
An interpolation routine is being used for intermediate values of the deadrise
angle between 12.5, 25 and 30 degrees.
A typical example of the goodness of fit of the polynomial approximation
compared with measured data of a model not belonging to the systematic series is shown in Figure 4. 0_3 2.2 -0.2 [ -DRCGExp. tot.
I /
/1
0.3 I.' 2.4 2.3 FriFigure 4 Measured and approximated sinkage and trim for an arbitrary
CD,C = 1.33
The resulting lift force distribution over the length of the model is shown in
Figure 5 for one particular situation. It is obvious that the dynamic lift prevails in the forepart of the wetted length of the planing ship and the hydrostatic part aft. The lift force due to the cross flow drag is of minor importance.
Force (Nfm) 500 400 300 2i Table 1. Model 251 ka 0.8526 30° abf 0.6395 Model 277 ka 0.8473 250 abf 0.7206 Model 276 ka 1.2658 12.50 abf 0.6239 Fastship Zarnick Ordinate 0 2 4 6 6 10 12 14 16 18 20
Figure 5 Lift force distribution over the length of the model.
According to Shufold [7], the cross flow drag coefficient of a section with
deadrise /3 may be approximated by:
CD,c = 1.33 cos/3 (10)
If the steady state still water equilibrium position is known as is assumed now by making use of the derived polynomials, the remaining unknown coefficients abf and ka may be determined by solving the two equations of motions
simulta-neously.
This procedure has been used on a variety of planing boat designs of which
model test data were available to yield values of a11,f and ka. The results for the parent models of the systematic series are presented in the Table 1.
From the formulations used for the dynamic lift it is obvious that the change of
sectional added mass over the length plays a predominant role. Since in this
calculation procedure the added mass is being related to the wetted beam of the sections and the beam is generally reducing in the aft body of a planing hull, a "negative" dynamic lift may occur in the aft body. This trend is not confirmed
by data from literature. As stated by Payne [12] by using a different
formula-tion for the added mass of the secformula-tions with a wetted chine this phenomenon may be eliminated. In the present formulation the negative slope of the added mass is neglected if it occurs and the hydrodynamic lift set to zero for these
sections.
2.2 Added Mass
As may be seen from the equations derived for the lift forces on the planing
hull, the added mass of the sections plays an important role, since the
derivati-ve of the added mass oderivati-ver the length of the hull determines the magnitude of the dynamic lift to a large extend. In the presented computational model the added mass is considered to be a function of the momentaneous wetted beam of the section using the Wagner formulation corrected for the pile up of the
water according to:
ma = ka2fpb2
in which ka is a constant for all sections.
Other methods to calculate the added mass and damping distribution over the
length of a hull moving at high forward speed are not available yet. So for the
calculation of these hydrodynamic reaction forces in those conditions use is
generally being made of a strip theory approach.
To check on the validity of this approach a series of oscillation experiments with a segmented model of the parent of the High Speed Displacement Hull
Form series has been carried out by Keuning [13] at the Deift Ship Hydrome-chanics Laboratory. This HSDHF series has been tested extensively at MARIN on resistance and motions in head waves in a research project co-sponsored by the Royal Netherlands Navy, the Royal Australian Navy and the David Taylor
Research Centre (USA). Using these results Blok and Beukelman [14] found
quite good correlation between measured and calculated heave and pitch
response operators of these models even at high forward speeds (i.e. Fn =1.14) using a linear strip theory calculation method.
From the study presented in Reference 13., it may be concluded that the
experiment and the calculations was important for improving the agreement between the measured and calculated data for the added mass.
Beyond this, the measured lift force distribution over the length of the fast
moving model in its proper reference position with respect to sinkage and trim
has been brought into the procedure used for the determination of the added
mass from the measured in-phase forces on the segments of the model. The lift
force distribution over the length of the model due to the forward speed and
the change herein due to the change of submergence of each section during the oscillatory motion has been measured in a quasi steady way. The time histories
of the force transducers of each section have been elaborated using these data
to yield the added mass of each section. The derived results show that the
added mass of the sections, using this method, becomes almost independent of the forward speed and the frequency of oscillation, ranging in that experiment
from o, = 4 to w = 15 for a two meter long model.
A typical result of these measurements is shown in Figure 6.
The results of these measurements justify the use of the frequency independent formulation of Wagner for the added mass.
Trim .l.62°
Based on actual restorin force
50
0
-25
50
0
Figure 6
Added mass distribution along the length of a model at high
forward speed. 5 - 7 w 13 1 2 3 5 6 7 -25 V.riton I + V.rai on 2 Mva%ur.AlQflt
By using this frequency independent formulation for the added mass, which is only based on the waterline beam of the section, it becomes easily possible to
make the added mass time dependent. In the formulation the momentaneous waterline beam of the section based on the relative motion is taken into the calculations. Hereto use is being made of a complete geometry description of
the ship from baseline to deckline.
To illustrate the relative importance of the time dependency of the added mass,
figure 7 shows the response operator for the vertical bow accelerations on the
25 degrees deadrise parent model calculated with and without the time depen-dency of the added mass.
acceleration/g 4 3 2 I 0 ('1
'4- Bow acc (time dep.) Bow acc (Fixed) Cc3 acc (time dep.) CG acc (Fixed) 0 Model 276 nAoI,,,.W.. (0 h 0 0 0 .0
Wave frequency (rad/s)
Figure 7 Response operator bow vertical accelerations with and without
time dependent added mass.
2.3 Wave Forces.
A similar approach has been followed with the wave exciting forces, another source of non-linear behaviour of a planing boat in waves. In assessing the
limits of the operability of a fast craft in waves, one is more interested in
extreme motions and peak values of accelerations than in the extrapolation ofbehaviour derived from otherwise linear calculations with infinite small
ampli-tudes.
0)
This implies that the large relative motions of the craft with respect to the
waves should be taken into account.
With the same segmented model
of the HSDHF parent wave force
measurements have been performed at high speeds, i.e. Fn = 0.57 and Fn = 1.14. The results of these measurements showed that the wave forces weredominated by the Froude Kriloff component. This enables the introduction of the nonlinear parts of these forces into the calculation procedure by using the
actual momentaneous submerged volume of the craft, due to heave, pitch and
wave elevation, for the determination of the Froude Kriloff force. In this
procedure also the geometry of the craft from baseline to deckline is beingused.
Zarnick in his method turned to vertical sidewalls above the chine. The
influence of this on the calculated results for the 25.0 degrees deadrise modelis shown in Figure 8. It should be noted that this influence extends in both the wave forces and the added mass calculation (and therefore the lift).
Acc./g 2 1 0 Non-prismatic Prismatic
Wave Frequency (rad/s)
Figure 8 Bow accelerations with and without vertical sidewalls above the
chine.
It is important to note that the buoyancy correction coefficient a1,f is also used in the determination of the Froude Kriloff forces on the ship in waves. This is an arbitrary method but based on the assumption that the character of the flow
around the hull, i.e. flow separation at the chine and transom and a stagnation line with spray area, on which the correction factor is based, remains roughly unchanged. From correlation with experimental results this appeared to yield
quite satisfactory results.
7
6
2 3 4 5
1
3
VALIDATION
The computational model has been validated extensively using the data of mo-del experiments carried out at the large towing tank of the Deift Ship
Hydro-mechanics Laboratory. The models used in the experiments were the three parent models of the systematic series of planing hull forms, as depicted in
Figure 3. The models have been tested in three different wave spectra of which the particulars are presented in table 3
Table 3
The ship length is 15 meters.
The speeds used during the experiments corresponded to a volumetric Froude number of 1.65 and 2.70 respectively.
The heave, pitch and vertical accelerations at the centre of gravity and the bow have been measured and elaborated in the usual way to yield significant values and maximum values.
The results of the calculated and measured vertical bow accelerations are
presented in Figure 9 on a basis of deadrise angle at midship section.
Specium I mcrcd sign if iC3fl t SpccCrurs msur:dT Spectrum 1 H113 = 0.55 m Tp = 5.9 sec Spectrum 2 H13 = 1.10 m Tp = 6.7 sec Spectrum 3 H13 = 1.60 m
Tp = 9.0 sec
0 00 20 20 30 30 DeadnseFigure 9 Measured and calculated significant and maximum vertical bow accelerations as function of deadrise.
Generally it was concluded from this validation study that the agreement
between the measured and calculated values for heave pitch and vertical
accelerations was in good agreement, although discrepancies still do occur.
4
OPERABILITY ANALYSIS
One particular aspect of Figure 10 should be emphasised a little more.
As may be seen from this figure the relation between the significant and the maximum value of the vertical bow acceleration is strongly dependent on the
deadrise angle at ordinate 10 (midship) but rather on the deadrise of the whole planing bottom, since all three models used were derived from the same parent
and the distribution of the deadrise over the length of the ship is similar in a
non-dimensional way. This implies that there is a considerable discrepancy between the predicted maximum value of the vertical acceleration in an irre-gular sea using a non-linear or a linear theory. For the high deadrise craft the relation between significant and peak value is roughly a factor of 2, while for
the low deadrise hull it may increase to 4 or 5.
Using linear theory and assuming the wave heights to be Rayleigh distributed, which is generally accepted assumption in ocean wave statistics, the maximum (or rather the 1/1000 wave) is approximately 2 times the significant value, both
for the wave and the vertical acceleration. From calculations it appears that a
linear strip theory motion calculation program is capable to predict the signifi-cant values of heave and pitch of a planing craft in a moderate wave condition
quite reasonable. The extrapolation to maximum values however may lead to
seriously underprediction of the peak accelerations.
A non-linear theory like the one here presented is capable of predicting the
peak values quite reasonable using the same wave spectrum as an input.
So, if the operability of a planing craft is under consideration,
it dependslargely on the kind of limiting criteria that are being used, what the outcome
will be:
Most commonly the limiting criteria are based on significant values of for
instance vertical accelerations, because these are rather easy to measure and to
calculate. In particular say a decade ago, no non-linear computational models were available to designers for assessing the motions of planing craft in
irre-gular waves. The underlying assumption was that the maximum values would
be twice the significant ones (for all craft similar) and these would therefore not exceed certain values. From full scale experiments on coastal patrol boats at the Dutch North Sea coastal waters however, it was concluded that rather the occurrence of one peak value was the limiting criterium for the voluntary
speed reduction by the crew and not the significant value.
This may lead to a different outcome of an optimisation as is shown by the results presented in Figure 10, in which the results of an operability analysis
been increased and decreased by 10% and the effect on the operability is
calculated using a scatter diagram from the North Sea and the same limitingcriteria. It should be noted that through the change in beam while maintaining the depth a considerable change in deadrise does occur.
100 L 80 60 >-a. 0 20 0
- lINEAR -e-- FA5T5I4P SIGNIF.
-9-- FASISHIP NAAIN.
5 5.5
BEAM
Figure 10
Operability of a planing hull with varying beam using both
linear and non-linear calculations.
It is obvious from this figure that the trend found from the beam variation on the operability using the non-linear program FASTSHIP is opposite from the
trend found from the linear calculations.
REFERENCES
Zarnick, E.E.; A non-linear mathematical model of motions of a
planing boat in regular waves. DTNSRDC-report 78/032 March 1978
Keuning, J.A.; Non linear mathematical model for the heaving and
pitching of planing boats in irregular waves. Ship Hydromechanics Laboratory, TU Delft.
Martin, M.; Theoretical Prediction of motions of high-speed planing
boats. Journal of Ship Research, Vol 22, No.3, Sept. 1978
Beukelman, W.; Prediction of operability of fast semi planing vessels in a seaway. Report no 700, Ship Hydromechanics laboratory.
[51 Sottorf, W.; Experiments with planing surfaces .NACA TM No.739,
1932
Sedov, L.; Scale effect and optimum relations for sea surface planing.
NACA TM No. 1097, Feb. 1947
Shufold,
Ch.L.; A theoretical and experimental study of planing
surfaces including effects of cross section and plan form. NACA report no. 1355, 1957
[81 Savitsky, D.; Hydrodynamic design of planing hulls. Marine
Technolo-gy, Vol.1, No.1, Oct. 1964
[911 Clement, E.P. and Blount, D.L.; Resistance tests of a systematic series
of planing hull forms. Trans. SNAME, Vol.71, 1963
Keuning, J.A. and Gerritsma,
J.; Resistance tests with a series of
planing hull forms with 25 degrees deadrise. International Shipbuilding Progress, Vol. 410, Sept. 1982
Keuning, J.A., Gerritsma, J. and van Terwisga, P.; Resistance tests
with a series of planing hull forms with 30 degrees deadrise. To be
Published.
Payne, P.R.; On the high speed porpoising instability of a prismatic
hull. Journal of Ship Research, Vol.28, No.2, June 1984.
Keuning, J.A.; Distribution of added mass and damping along the
length of a ship model moving at high forward speed. InternationalShipbuilding Progress, Vol. 37, no. 410, Sept. 1990
Blok,
