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OCR for page 141
A Stochastic Analysis of Nonlinear Rolling
in a Narrow Band Sea
A. Francescutto and R. Nabergoj (University of Trieste, Italy)
ABSTRACT
The extensive analysis carried out i n last
years on the nonlinear rolling of a ship, through the
use of different versions of perturbation methods,
allowed many researchers to discover some very
peculiar features of this motion. In a deterministic
beam sea, it appeared the possibility of
resonances different from synchronism, in
particular subharmonic oscillations in both the
upright and heeled conditions. Moreover, the
resonance peaks as a function of tuning ratio are
bent, mainly as a consequence of righting arm
nonli nearity, so that i n a suitable range of
frequencies the oscillation can have three steady
states of very different amplitude.
Later on, the possibility of successive
bifurcations to chaotic behaviour was detected and
it is actually widely and deeply studied. This
"deterministic chaos" is in fashion and is very
interesting in principle; on the other hand it is still
related to the appearance of a strictly mono-
chromatic sea, which is a possibility that can be
hardly realized in a real seaway.
At the other extremum, stochastic nonlinear
rolling was treated mainly as a process with
sufficiently broad band to have no possibility of
jump phenomena and no research was carried out
taking the bandwidth as a parameter. To clarify this
puzzling aspect, the nonlinear model introduced in
previous work with the excitation represented by
white noise filtered through a linear filter was
studied in detail. The filter was allowed to vary
centerpeak and bandwidth of the excitation. A
representation as a sum of two sinusoidal
processes with Gaussian slowly varying
amplitudes was introduced. This description is
particularly suitable for description of narrow band
processes. The system of differential equations
rolling motion plus filter, was solved by means of
the perturbation method of multiple scales. Several
curves roll variance/tuning ratio were obtained
varying the bandwidth for an excitation variance
corresponding to a moderate sea. The results
i ndicate that, provided the bandwidth of the
141
excitation is not greater than a sharp JONSWAP
spectrum, at least in the considered cases, the
main resonance exhibits the same characteristic
features shown in the deterministic, i.e.
multivaluedness and jump possibility. This
similarity is related to the fact that narrow
bandwidth means high autocorrelation.
The perturbation method allows to obtain
simplified equations giving the maximum roll
variance as a function of the excitation variance.
These results are confirmed through the
envelope analysis of a preliminary stochastic time-
domain numerical simulation. The curve giving the
probability density as a function of the variance of
the motion, appears, in fact bimodal. Finally, the
region of the first subharmonic has been analysed
giving the response curves for the roll variance of
the subharmonic component and the excitation
variance threshold for its onset.
INTRODUCTION
The connection between large amplitude
rolling and capsizing is not very clear. In principle,
the large amplitude rolling motion of a ship can be
a stable motion, provided it be included in some
stability boundary. In practice, the ship is a very
complicate system, so that many dramatic
scenarios can appear once large amplitude rolling
is in some way originated. The reasons of thiese
practical differencies can be summarized as
depending on:
- wrong operation of the ship. This is for
example the case when the rudder is released in
the attempt to recover from an excessive heeling in
manoeuvering, non properly adjusted antirolling
tanks are employed to reduce excessive rolling, a
route change is adopted to avoid the beam or
quartering action of wave trains;
- structural failure with consequent opening of
holes in the hull and flooding of some
compartment;
- effect of additional heeling moments due to
water loading on deck and through the deck
openings caused by the large amplitude rolling or
OCR for page 142
shifting of cargo due to the large rolling
accelerations. This last phenomenon can in turn
give raise to a positive feedback with final
. .
capsizing or originate a structural failure
- and of course, the concurrency of more than
one of this phenomena. The analysis of casualties
at sea reveals that this is often what happens when
a ship is lost.
Actually, it is not very simple to quantify the
relative i mportance of these consequencies of
large amplitude rolling. On the other hand, neither
the probability of large amplitude rolling has been
stated in a satisfactory way. This paper is thus
devoted to a first analysis of this problem.
In previous papers [1-4], it has been shown
that large amplitude rolling can be a consequence
of a jump between the antiresonant state and the
resonant one, or between non resonant and
subharmonic states, due to the strong deviation
from linearity of the rolling motion. This originates
the possibily of bifurcations that can be effective as
a consequence of some change in the parameters
in the case of a deterministic excitation, and due to
the very nature of the oscillation when a narrow
band stochastic excitation is considered.
The phenomenon is not very fast, but quite
difficult to forecast, inasmuch its occurrence is not
related to some dramatic change in the excitation
statistics. Moreover, the amplitude differences can
be dramatic also in presence of non very intense
excitation. Therefore, it appears of interest a
parametric research to find the probability of its
occurrence. To this end, the probability density
function of the response levels should be
computed by means of approximate analytical
methods in the following three cases of interest:
- bifurcation between the anti- and the
resonant oscillation in the main resonance region
in stochastic beam sea;
- bifurcation between non resonant and
subharmonic in the first subharmonic region in
stochastic beam sea;
- bifurcation between zero amplitude (or
negligible one) and parametric suharmonic rolling
in stochastic following or quartering sea.
As a consequence of the possibility of
bifurcations, the probability density functions (pdfl
can be bimodal functions for some values of the
parameters. This allows an evaluation of the
probabilities of the considered different oscillation
states, and in particular of the probability of the
onset of large amplitude rolling.
In this paper, we will devote our attention to
ship rolling in a beam sea, with particular regard to
the possibilites offered by the bifurcations in the
regions of synchronism and of the main
subharmonic. The analysis is limited to the
obtaining of the frequency response curves in
terms of roll variance in synchronism and to the
frequency response and excitation threshold in the
case of subharmonic.
NARROW BAND EXCITATION
The inclining moment due to the action of the
sea waves, is usually represented by means of the
introduction of the energy spectrum of the sea.
Extensive campaigns of measurements allowed to
obtain a statistical description of the different
regions of the world. From these tables, some
standard descriptions have been obtained, as the
ITTC and ISSC standard spectra, and successive
corrections and integrations. These spectra are
usually considered to be sufficiently narrow band
to allow a simplified statistical treatment, and at the
same time sufficiently broad band with respect to
the system transfer function to allow a simplified
treatment of the response [5,6~. In addition,
hypoteses such as Gaussianity, stationarity and
ergodicity are often assumed to further simplify the
analysis. For a detailed discussion of these
problems see Ref. [7] .Actually, this procedure is a
Ratifying one when quasi-linear and non extreme
phenomena are analysed. If we devote our
attention to ship rolling, this is unfortunately not
true, since it is a highly nonlinear, large amplitude
motion, that can lead to extreme phenomena such
as capsizing. It is not the scope of this paper to
investigate in detail the connection between large
amplitude rolling and capsizing. We will assume
that large amplitude rolling is in itself an extremely
interesting topic and we devote our attention to
some conditions, non sufficiently investigated until
now, that can lead to large amplitude rolling.
As shown in previous papers [1-4] in the
deterministic domain, the rolling motion, due to its
inherently high degree of non linearity, can exhibit
peculiar phenomena leading to non expected
large amplitude oscillations, also in weak to
moderate seas. This is due to the appearance of
different steady states of oscillation corresponding
to the same excitation intensity and the consequent
possibility of bifurcation with jumps among these
different states. In the deterministic approach, the
problem is an initial value problem, so that the
phase plane, or the Van der Pol plane are divided
in different regions, called domains of attraction,
each leading to one of the steady state solutions.
Neglecting the appearance of chaotic behaviour,
that seems not playing a dramatic role, the jump is
thus only possible when there is a change in the
parameters defining the system or the excitation,
such as cargo shifting, water on deck, wind gusts,
change of heading.
The build-up of a jump is not an immediate
process, requiring a few cycles of the excitation to
be completed. In a very broad band stochastic
excitation case, it is thus very unlikely to occurr due
to the very poor correlation among successive
cycles. Actually this only means that its probability
is very low, i.e. it is a 'rare event'.
On the other hand, ITTC spectrum is only an
average one, so that its parameters are subject to
statistical uncertainty. Moreover, a peak magni-
fication factor has been suggested for the
description of North Atlantic. With this correction,
142
OCR for page 143
one gets the quite sharp JONSWAP spectrum.
The description of the sea action through the
spectrum, is a short term description, valid for the
well developed sea. It does not exclude the
possibility of appearance of narrow and very
narrow spectra in particular conditions, as for
example crossing channels or in port regions or in
the developing zone, or simply as a statistical
fluctuation of broader spectra.
In this case, there could be sufficient
correlation among cycles of the excitation to allow
for the jump between different amplitude states,
without the need of any change in the system
parameters.
The spectrum of the sea excitation f will be
described through a simple linear filter of the type:
Sf = ~co02SO/~(cof2-~2~2+c,~2~,2] (1 )
shaping Gaussian white noise W of level SO The
excitation is zero mean and has variance of2=~So.
It is suited to describe narrow band excitation, as in
the limit for y - 0 it reduces to the sinusoidal
excitation
f = ewcoscoft , (2)
with eW2=2of2, whose spectrum is represented by
6(o)
In the following, the perturbation method of
the multiple scales will be adopted. For this, it is
convenient to represent the narrow band process f
in the Stratonovich form [8~:
fits = Fc~t~cos~ft + Fs~t~sinc3t
where Fc and Fs are slowly varying independent
Gaussian processes represented by the equations:
dt = (2) WC- (2 - ) Fc
dt = (2) Ws~ (27) Fs
where We and Ws are independent Gaussian
white noise processes with the same spectrum
level of W. For further details see Ref. [9,103.
Actual values of the spectrum level SO,
bandwidth parameter ~ and centerpeak frequency
of can be obtained through a nonlinear fitting of the
spectra to be approximated.
This procedure applied to normalized
standard wave height spectra Sly using IMSL
library routine RNLIN, provided the following
values [1 13:
- ITTC
So=.35 cof-1.16 ~=.50 is= 0
143
- JONSWAP with sharpness magnification factor
equal to 7 (maximum value)
So=.13 c~=1.01 y=.14
The transformation of wave heights into
inclining moments actually broadens a little the
spectrum. In any case, it appears that, also in the
frame of standard spectra, there is the possibility of
effectively narrow band spectra, in the sense of
. . . . .
giving rise to Jumps in ro Ing.
In the following, a value for If Will be used so
as to give an excitation intensity corresponding to
ew =.2, a value used in previous calculations in the
deterministic domain. This value represents a
sinusoidal wave with effective wave slope well
below the limit for breaking waves. The bandwidth
factor ~ will be varied as a parameter.
THE EQUATION OF MOTION
In the light of the preceding discussion, the
rolling motion in beam sea will be represented by
the following nondimensional nonlinear differential
equation:
ddt2 + (2~+81 x2) dt + 62 (d-t)
+ coO2x + a3x3 = fits (3)
We do not discuss here the meaning of the
different terms, that can be found in Ref. [13.
The excitation f is represented by the
expression (2) in the deterministic case, whereas
in the stochastic it is represented by the solution of
the following differential equation:
d2f +,~,df + c'~f2f = ~/2mfW (4)
As mentioned, an approximate solution i n
terms of the variance O2 of the motion will be
obtained in the synchronism region Cof _ Coo and in
that of the main subharmonic Of ~ 3c,)O.
To get some insight into the nature of the
solutions, attention was preliminarly focussed on a
form of Eq. 3 containing the minimum of
nonlinearities, i.e. only the righting arm one. In
these conditions, the rolling motion is described by
an equation of the Duffing type. This is tied to the
fact that, in the deterministic approach, it is this
nonlinearity that is responsible of the presence of
bifurcations, whereas the damping moment
nonlinearities play an important role in determining
the maximum values that can be reached by the
oscillation amplitude.
Once obtained a solution valid in the
stochastic narrow band case, the deterministic
sinusoidal solution can be recovered by means of
the following positions:
OCR for page 144
c2
,s2= 2
e 2
~ W
tuft = 2
2 Q2
6Q = 2
(5)
being C the amplitude of the nonlinear
response in resonance and Q the amplitude of the
linear response out of resonance, both in the
deterministic case
Only the results relative to the application of
the method of multiple scales [12] will be reported.
SYNCHRONISM REGION
The rolling motion in the region of
synchronism is statistically governed by the
following equation:
ja32(S6 + 3a3(tI)o2-cof2~4 + [(t,)o2 t~f2~2 +
(2~fJ2~1 +/2 = (1 +~/2~) of 2 (6)
that, using the positions represented by Eq's 5,
reduces to:
a
0.5
~0.0
~ ~5
o.o~ , 1
0)f/Ct)o2.0
1.0 1.5
Fig. 1 Roll variance ~ as a function of tuning
ratio of/mot The following values have been
used for the parameters: oc3=4.0, lo=. 05,
rs~=.1414 (eW=.2~. The number on the curves
indicates the value of y.
~OC32C6 + ~X3(COo2-COf2)C4 + 0-5
[(COo2-Wf2~2 + (2~2]C2 = eW2 (7)
in the sinusoidal case. Eq. 7 agrees with previous
results [1 ].
Eq. 6 represents a very important result
inasmuch as it allows a relatively simple
calculation of the frequency response curve in
presence of stochastic excitation. Due to the
particular way through which it was obtained, its
validity is limited to the narrow band case.
In Fig. 1 and Fig. 2, the response curves
given by Eq. 6 are reported as a function of tuning
ratio centerpeak frequency/natural frequency of
small amplitude oscillations too for different values
of the bandwidth parameter including the limiting
case of sinusoidal excitation.
Fig. 1 represents a tipical case of modern
containership, whereas Fig. 2 refer to a more
traditional case [13~.
As one can see, the possibility of bifurcation
is preserved also in the stochastic case, provided
that the bandwidth is sufficiently small. In
particular, in the considered cases, the most sharp
JONSWAP is not far from the higher limit of values
of That gives multiple regime of oscillations.
In the sinusoidal case, in the frequency
region where three states are possible, only the
two extreme represent stable oscillations and the
bifurcation involves them. In the stochastic case,
~ 0 Of/mo
Fig. 2 Roll variance ~ as a function of tuning
ratio Cof/Coo. The following values have been
used for the parameters: a3=-0.5., ,u=.05,
Of=. 1414 (eW=.2~. The number on the curves
indicates the value of y.
the concept of stability of an oscillation state looses
part of its meaning. Actually, in the frequency
region where three levels of variance are possible,
the roll variance oscillates, having local maxima of
probability in correspondence to the two extreme
values. This is intrinsically connected with the
nature of the stochastic excitation [143.
Neglecting the contribution of the nonlinear
term, the solution far from resonances is obtained
as:
OCR for page 145
aQ2 =(1 +~y/2~) c~f2 / [(COo2 c,3f2)2 +
(2pc'Jf)2(1 +~/2t,)2] _ tI~2/(mo2-mf2)2 (8)
A perturbative analysis on Eq. 6, considering
r,)O+of_ Woo gives a second degree algebraic
equation on the variable Ace = Cof-mo. The reality
condition of the solution gives a simple formula
relating the maximum value of the roll variace am2
to the excitation variance:
am2 =CIf2 / 4,u2cl)o2~1+~/2~)
that reduces to the sinusoidal expression: or
Cm2 =eW2 / 4~2coo2
These expressions correspond to a linear
approach and thus generally they are not
sufficiently reliable.
A corresponding analysis conducted on the
complete nonlinear rolling Eq. 3, allowed us to
obtain the general equation for the roll variance in
the implicit form:
of2 = ~/2 [(Oo2 cl)f2~2 - ~a362
method of multiple scales permitted to obtain an
equation describing the variance case of the
resonant part of the solution, i.e. the amplitude of
the subharmonic component that has to be added,
in some way, to the non resonant component of
variance aQ2. Here, only the results relative to the
Duffing case will be presented. The analysis of the
full Eq. 3 is actually in progress.
As a consequence of the stability of the non
resonant component, we have the following
possibilities:
6S2 = 0
;~32~}s4 + [3a3Qn-~a32~1 +~y/6~1)CJQ2 ]aS2
+[Qn2+~2~2~1+~/6l,l,)21 _ O ~10)
with
Qn = C)02-mf2/9 + 3a3~,Q2
Eq. (10) is a second degree algebraic
+ (2cl)o~eq)2~1~l2peq)2]cs2 equation in the variable css2. The reality condition
with
Req - ~ + 4 O2 ~eq = (~, +3C)o262) 0.4
A perturbation on this equation gives the
implicit equation for the maximum roll variances: 0
(2moJ2~2~1 +~y/2Req)om2+~L(1 +~/2~eq) 0.2
~eq6m4+~eq2Csm6] = (Sf2 `9'
the corresponding sinusoidal expression proved
good in the forecasting of the maximum roll
amplitude in synchronism [153. Here, the
predictions are, of course, of a statistical nature,
but sufficient for many practical scopes. When a
bifurcation is possible, Eq. 9 refers to the highest
variance level, irrespectively of its actual
probability of occurrence..
SUBHARMONIC SOLUTION
In the frequency region COf_ 3mo the non
resonant solution given by Eq. 8 is generally
stable. Nevertheless, the analysis conducted in the
sinusoidal case, indicated that there is the
possibility of excitation of a resonant oscillation
with frequency of/3 _ mo The application of the
o.,
ao
,
{th
.
3.0 35 Of/Oo 40
Fig. 3 Variance os of the subharmonic
component, excitation threshold CIfth for
subharmonic oscillation onset and variance aQ
of the out of resonance component as a
function of tuning ratio cl)f/cOo. The following
values have been used for the parameters:
a3=4.0., p=.005, of=.1414, ~.01.
145
OCR for page 146
for the solution gives the threshold value ofth2 Of
the excitation variance for the onset of a
subharmonic component of the oscillation:
with = - 3a (1+~/21l) [~mo2 c,~f2~2 +
(2~2~1+~/2~2]{Wo2-~2l9 + ~ 3
t~ 2 2/9~2 4 2 2~1 /6
(7-~16~] ~ /2 } /~7-~/6~)
This expression also reduces to the previous
one [1 ~ i n the sinusoidal li mitt
In Fig. 3 and Fig. 4, the values Of `ss, 6Q, and
fifth are reported as a function of centerpeak
0.0
Fig. 4 Variance as of the subharmonic
component, excitation threshold Fifth for
subharmonic oscillation onset and variance aQ
of the out of resonance component as a
function of tuning ratio Of/Coo. The following
values have been used for the parameters:
a3=-1.75., p=.005, 6f=.1414, y=.01.
frequency of the excitation. As one can see, the
excitation of a subharmonic component is a
phenomenon possible also in the stochastic case.
The required bandwidth is in general quite narrow
and the roll damping quite low, so reducing the
probability of occurrence.
NUMERICAL SIMULATION AND CONCLUSIONS
Numerical time domain simulation in the
stochastic excitation case is not, in general, an
easy task. This is due to a lot of complications
arising in the realization of a true stochastic time
series for the excitation as explained in [73. In
addition, is not clear what happens of the domains
of attraction that define the set of initial conditions
that lead to a particular steady state solution. On
the other hand, in the limit of narrow band the
system has to recover the sinusoidal features.
Looking forward to the possibility of
presenting a more complete numerical picture of
the behaviour of this nonlinear system in stochastic
domain in next future [14], we report here only
some conclusions obtained from preliminary
calculations.
Time domain simulation of Eq. 3 with the
forcing term given by the following expression [163:
fits = ( N ) TCOS(okt+Ēk)
This gives a pseudorandom signal when the
COk are chosen independently from a random
population with a pdf of the same form of the
spectrum of f and the phases Ok are indipendent
and uniformly distributed in (0,2~.
A time domain solution was obtained in the
synchronism region where multiple regime was
indicated. A statistical analysis on the time series
indicated a bimodal pdf.
These results confirm that jump possibility is
likely to occurr in the domain of parameters
indicated by approximate perturbative solutions.
The probability of the highest state depends on the
tuning ratio, decreasing as this increasingly
deviates from the synchronism condition.
ACKNOWLEDGEMENTS
The authors would expresse their gratitude to
Prof. H. G. Davies for the interesting discussion on
narrow band processes during his visit in Trieste.
REFERENCES
[1] Cardo, A., Francescutto, A., Nabergoj, R.,
"Ultraharmonics and Subharmonics in the Rolling
Motion of a Ship: Steady-State Solution",
International Shipbilding Progress Vol. 28, 1981,
pp. 234-251.
[2] Cardo, A., Francescutto, A., Nabergoj, R.,
"Deterministic Nonlinear Rolling: A Critical
Review", Bulletin de ['Association Tec h n i q u e
Maritime ed Aeronautique, Vol. 85, 1985, pp. 119-
141.
[3] Cardo, A., Francescutto, A., Nabergoj, R.,
"Transient Nonlinear Rolling: The Domains of
Attraction", Proceedings 14th Scientific and
Methodological Seminar on Ship Hydrodynamics,
Varna, Vol. 2, 1985, pp. 23-29.
[4] Francescutto, A., '~Jump phenomena in
Nonlinear Rolling in a Stochastic Beam Sea",
Bulletin de ['Association Technique Maritime ed
Aeronautique, Vol. 88, 1988, pp. 504-524.
[5] Roberts, J. B., "A Stochastic Theory for
Nonlinear Ship Rolling in Irregular Seas", Journal
of Ship Research, Vol. 26, 1982, pp. 229-245.
[6] Cardo, A., Francescutto, A., Nabergoj, R.,
146
OCR for page 147
"Stochastic Nonlinear Rolling: Which Approach?",
Bulletin de ['Association Technique Maritime ed
Aeronautique, Vol. 87, 1987, pp. 491-505.
[7] Francescutto, A., "On the Nonlinear Motions of
Ships and Structure in Narrow Band Sea",
Proceedings of the IUTAM Symposium o n
Dynamics of Marine Vehicles and Structures in
Waves, London 1990
[8] Stratonovich, R. L., Topics in the Theory of
Random Noise' Vol. 1, Gordon and Breach, New
York, 1963.
[9] Rajan, S., Davies, H. G., "Multiple Time Scaling
of the Response of a Duffing Oscillator to Narrw-
Band Random Excitation", Journal of Sound and
Vibration, Vol. 123, 1988, pp. 497-506.
[10] Davies, H., G., Rajan, S., "Random
Superharmonic and Subharmonic Response:
Multiple Time Scaling of a Duffing Oscillator",
Journal of Sound and Vibration, Vol. 126, 1988,
pp. 195-208.
[11] Francescutto, A., Cardo, A., Contento, G., "On
the Representation of Sea Spectra Through Linear
Filters" (in Italians, University of Trieste, Institute of
Naval Architecture, Technical Report N. 67, 1988.
[12] Nayfeh, A. H., Mock, D. T., Nonlinear
Oscillations, Wiley-lnterscience, New York, 1 979.
t13] Cardo, A., Francescutto, A., Nabergoj, R., "The
Excitation Threshold and the Onset of
Subharmonic Oscillations in Nonlinear Rolling",
International Shipbilding Progress,Vol. 32, 1985,
pp. 210-214.
[14] Francescutto, A., "On the Probability of Large
Amplitude Rolling and Capsizing as a
Consequence of Bifurcations", Accepted for
presentation at the 10th International Conference
on Offshore Mechanics and Arctic Engineering
'OMAE', Stavanger, June 1991.
[15] Cardo, A., Francescutto, A., Nabergoj, R.,
"Nonlinear Rolling in Regular Sea", International
Shipbilding Progress,Vol. 31, 1984, pp. 3-7.
t16] Shinozuka, M., "Simulation of Multivariate and
Multidimensional Random Processes", Journal of
Acoustical Society of America, Vol. 49, 1971, pp.
357-367.
DISCUSSION
All H. Nayfeh
Virginia Polytechnic Institute and State University, USA
How realistic are the results presented in Fieures 1 and 3 for the care
of a hardening-type righting arm? What are the assumptions
underlying the derivation of egs. (6), (7), (8), (9), and (10)? Are
they the same as those used by Davis and Raj an? How different are
your results from their results? What are the limitations of the
method?
AUTHORS' REPLY
The validity of the procedure adopted to get the perturbative
approximate solutions is implicitly proved by the goodness of the
comparison with the results of a numerical simulation (see [14] for
more details). As regards the meaning of considering the case with
hardening stiffness (`x370), we can deserve the following. A research
on a consistent specimen of ships in a variety of loading conditions
[ 17], indicated that the validity of a model of righting arm with ~3 > 0
can extend to quite large amplitude inclinations (30° - 40°) especially
for modern container ship forms. This is sufficient to assess the
importance of the inclusion of this case in a true nonlinear seakeeping
context, as far as rolling motion is considered. Capsizing, instead,
is generally too a complicated phenomenon to be explained by means
of relatively simplified models like those considered. On the other
hand, when large amplitude rolling is generated, it is likely to think
that additional mechanisms can deteriorate the stability qualities of the
ship, leading to capsize as indicated in the introduction (see also ref.
[14]). Additional references [17] Cardo, A., Francescuto, A.,
Nabergoj, R., "The Excitation Threshold and the Onset of
Subharmonic Oscillations in Abulinear Rolling, International
Shipbuilding Progress, Vol. 32, 1985, pp. 210-214.
DISCUSSION
Odd Faltinsen
Norwegian Institute of Technology, Norway
Is the frequency dependence of hydrodynamic coefficients included in
the model?
AllTHORS' REPLY
Actually, the frequency dependence of the hydrodynamic coefficients
is not included in the model. This is not due only to the obvious
overcomplications involved, but to the necessity of a better
clarification of the exact meaning of frequency in presence of a
stochastic excitation with the possibility of multiple regime of
oscillation and of the techniques for transfering this frequency
der~endence when doing time domain simulation with nonlinear
- -r
v
models. On the other hand, in a full nonlinear approach, it is not
very easy to separate added mass and damping contributions.
147
OCR for page 148
Representative terms from entire chapter:
narrow band
