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Representative terms from entire chapter:
quartering seas
Rolling of Biased Ships in Quartering Seas
N. Sanchez
(University of Texas at San Antonio, USA)
A. Nayfeh
(Virginia Polytechnic Institute and State University, USA)
ABSTRACT
The equation governing the nonlinear rolling of a
biased ship in quartering seas can be reduced to a
nonlinear orclinary-differential equation with parametric
and external 'excitations. The solutions of this equation
are analyzed by performing analog-computer
simulations to locate bifurcation points in a
two-climensional parameter space consisting of the
waveslope and encounter frequency. The predicted
instabilities for various levels of the parametric
excitation are summarized in bifurcation diagrams that
display the Stable regions and the instabilities that take
the ship into "dangeroust'responses.
INTRODUCTION
I n the study of the roll motion of a vessel it is
common to have an equation of motto', with
time-varying coefficients, which arise from
time-dependent restoring moments dire to the ship's
position on the wave (7,2) or to changes in the
displacement volume resulting from coupling with
other modes (3). These time-varying coefficients
constitute what is known as parametric excitations (4).
A considerable number of roll studies (e.y., 1-3, S-10)
analyzed the i'~fluenc.e of these excitations on the
stability of a ship in the presence of resonances, which
can lead to capsizing under rather mild sea conditions
(9,1 a).
In pre\/iot~s work (9,10), we applied an
analytical-nt~merical procedure to characterize the
stability of the steacly-state response of a ship when a
parameter is slowly varied. Using this procedure, we
analyzed the stability of an approximate analytical
solution by using Floquet theory and elementary
concepts of bifurcation theory. In (10), we have shown
that a ship model under a purely parametric excitation,
which is the case of rolling in longitudinal \vaves in the
absence of heeling moments and pitch-roll coupling,
displays self-similar behavior near the resonances.
Instabilities can lead to capsizing through two
scenario`,: one evolving from a large oscilla1 ion
through the disappearance of a chaotic attractor
(crises) and a second' potentially more
an explicit time dependence, which might come from
two sources (2): the position of the ship on the wave
or variations in the displacement volume due to heave
coupling. Win are mostly interested in the latter effect;
however, consideration of the first effect only changes
the nu merical values of the coefficients. The
righting-moment function is approximates! as (5,12)
K(0, t) = too td + a393 + a565 + he cost (2)
where coo is the linear undamped natural frequency of
the ship, the odd polynomial fits the ship's righting
moment curve, and the parametric term represents
heave - roll coupling expressed by the coefficient
K`'zaz
h=
2(oo
Here, Koz is the magnitude of the coupling coefficient
and az is the amplitude of the heaving motion, which is
assumed to be harmonic with frequency S2. The
damping moment D(~) is expressed as
D(~)-id a- q3B3
Assuming that the wavelength of the wave is large
compared with the ship's beam, we can write the
waveslope elf a regu lar beam sea as a harmonic
^
function or = Am COS fat, where Em is the maximum
waveslope. Using (1b)-(4), we rewrite (1a) as
+ 2,u~ ~ p3~73 ~ o'2 t~ + a303 -a ~505 + kit) cos(S2~]
= 070 its + (X305. + WAS] + ~ ~ 5' COS(~QT + ]~)
where y represents a phase angle between the wave
and heave motions and As is the bias angle produced
by the moment B.
l-o simplify the governing equation, we introduce
the time scaling t= (idols which transforms (5) into
B+2~-~3t}3- 0+a364-~505+h~cos(fit'
= As -a ~30S + ASKS + f, cos(S2t) ~ f2 sin(52t)
where tier, overdot re~,i-esents the derivative with
respect to t,
^ °`mli2
f1 = t1 ~ ~-1) IS y
and
amlQ
2 (/+,SI) n y
Using the transformation
= Bs + u
we rewrite (6) as
u+2pu+q3`i31- u+ bju+b2u2+b3u3
(7)
+ b4u4 + b5u5 -t hu cos(S2t) = f, cos(S2t)-f2 sil,(
The third attractor represented in the bifurcation
diagram in Figure 1 is the subharmonic response near
Q = 2, which is stable in the region between P4 and
J2. Figure 4 shows selected attractors in this region: (a)
shows the period-one response, (b) shows the
subharmonic resonant response obtained after
crossing P4 from right to left, (c) and (d) show
quantitative changes in the subharmonic response, (e)
shows the subharmonic response after a
period-doubling bifurcation across P2, and (f) shows a
chaotic attractor to the right of J2
Figure 5 shows the bifurcation diagram obtained
from an analog-computer simulation of (5) when the
parameters are set at V = 0, 49s= 1- 6°, and h = 0.3. In
this case the dotted region is much larger than that for
the case of an unbiased ship. The region of stability of
the subharmonic response shrank and the period-one
and subharmonic responses only coexist in a narrow
region between Q = 1.5 and 2.0. The curves S. and S2
represent saddle-node bifurcations of the large and
small attractors of the period-one resonant response,
which produce jumps in the corresponding attractors.
Figure 6 shows the two attractors for Q - 0.768 and
0Cm= 0.217. In Figure 5, S. represents a jump from the
large to the small attractor; S2 represents a jump in the
small attractor, which takes the system to the large
attractor if am is below tire black dot or to capsize if it
is above this point; and P.' represents period-doubling
bifurcations, leading to either chaos in the portion of
the curve enclosed by E' or directly to an instability in
the rest of the curve. The instability results in either
capsize when on, is above the black dot on P. or a jump
to the subharmonic attractor when it is below P..
Figure 7 shouts the coexisting attractors of the
period-one and subharmonic responses at Q -- 1.758
and 0Cm = 0.046 . The curve E, in Figure 5 represents the
point of capsize. For values of Q below the black dot
on E', a period-doubling sequence to chaos such as the
one shown in Figure 8 is observed. For crossings of
E, between the black dot and the point of merge with
P., a saddle-node bifurcation is observed, making the
period-2T solution unstable and capsizing takes place.
Figure 5 also shows that the stability region of the
subharmonic response is narrower than that in Figure
1. Crossing P3 from right to left causes the period-one
response to lose stability and the subharmonic
becomes stable. When P2 is crossed, the system
undergoes a sequence of period-doubling bifurcations
leading tie chaos, which disappears when E' is rear bed,
causing the ship to capsize. Figure 9 shows selected
phase portraits of the above changes in the solution:
(a) shows the period-one response, (b) and (c) show
the subharmonic response after crossing P3, (d) shows
a period-doubling in the subharmonic after crossing P2,
and (e) shows a second period doubling in the
subharmonic response.
Figure 10 shows the bifurcation diagram obtained
from an analog-computer simulation of (5) when the
parameters are set equal to: y -- 0, As--- 6°, and h
= 0.3. Figure 11 shows the attractors of the primary
resonance. Across curve S in Figure 10, the small
attractor undergoes a saddle-node bifurcation that
produces capsizing. Across P.' the large attractor
undergoes period-doubling bifurcations. In the portion
of P. enclosed by E' a sequence of period-doubling
bifurcations culminating in chaos is observed, in the
rest of the curve capsizing occurs after the first
period-doubling bifurcation. The large attractor also
undergoes period-doubling bifurcations across P4. The
subharmonic response undergoes period-doubling
bifurcations across P2 and P3. Capsizing is observed
after crossing P.; and E2. Figure 12 shows the coexisting
attractors that are found between P2 and P4.
CONCLUSIONS
The present resmelts show that the rlyna'~irs of a
positively biased ship is different from that of a
negatively biased sl~ip, which confirms the
experimental observations of Wright and l'/larshfield
(12). However, it is difficult to conclude which of the
two cases is more stable because the stability depends
on the region of the parameter space tinder
consideration. Nevertheless, it is clear that the motion
is very sensitive to bias and the region where capsizing
is observed grows considerably and appear to be
bigger for positively biased ships. VVe have also found
the bifurcation diagram to be sensitive to changes in
the phase angle y. The most interesting feature is the
major qualitative changes that the bifurcation diagrams
undergo as the bias angle changes. This clearly
suggests the need for a complete analysis before a
particular design can be considered safe to operate.
ACKNOWLEDGE/\fENT
This work was supported by the Office of Naval
Research under Contract Nos. N00014-83-K-0184/NR
43227 53 a n d N 0()0 1 4 -90-J - 1 149 ~
REFERENCES
Kerwin, J. E., "Notes on Rolling in Longitudinal
Waves," International Shipbuilding Progress, Vol
2,1955,pp.597-614.
Feat, G. and Jones, D., "Parametric Excitation and
the Stability of a Ship Subjected to a Steady Heeling
Moment," International Shipbuilding Progress,
Vol. 28, 1984, pi:). 263-267.
Paulling, J. R. and Rosenberg, R. M., "On Unsl-able
Ship Motions Resulting from Nonlinear Coupling,"
Journal of Ship Research, Vol. 3, 1959, pp. ~,6-46.
4. Nayfoh, A. H. and Mook, D. T., Nonlinear
Oscillations, Wiley-l nterscience, New York, 1979.
. Blocki, W., "Ship Safety in Connection with
Paramet'ic Resonance of the Roll," International
Shipb`'ildincl Pro'3ressq Vol. 27' 1980, pp 36-53
6. Abicht, V]., "On Capsizing of Ships in Regular and
Irregular Seas," Proceedings of the International
Conference on Stability of Ships and Ocean
Vehicles, Glasgow, 1975.
7.
135
Wellicorne, J. F., ",Nn Analytical Study of the
Mechanism of Capsizirig," Proceedings of the
International Conference on Stability of Ships and
Ocean Vehicles, Glasgow. 1975.
8. Skomedal, N. G., "Parametric Excitation of Roll
Motion and its Influence on Stability," Second
International Conference on Stability of Ships and
Ocean Vehicles, S111-3a, October 1982.
9. Nayfoh, A. H; and Sanchez, N. E., "Stability and
Complicatecl Responses of Ships in llegular Beam
Seas," to appear, International Shipbuilding
Progress., 1990.
10. Sanchez, N. E, and Nayfoh, A. H., "Nonlinear
Rolling Motions of Ships in Longitudinal Waves," to
appear, International Shipbuilding Progress, 1990.
11. Parlitz, U. and Lauterborn, W., "Superstructure in
the Bifurcation Set of the Buffing Equation,"
Physics l esters, Vol. 107A, 1985, pp. 351-355.
12. Wright, J. H. G. and Marshfield, W. B., "Ship Roll
Response and Capsize Behavior in Beam Seas,"
Transactions Royal Institution of Naval Architects,
Vol. 122, 1980, pp. 129-147.
13. Nayfoh, A. H., Introduction to Perturbation
Techniques, Wiley-1 nterscience, New York, 1981.
14. Berge, P., Pomeau, Y. and Vidal, C., Order Within
Chaos, Wiley-lnterscience, New York, 1984.
0.20
Of 0.1 5
0.05
0.00
- Tact
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Q
Figure 1. Bifurcation diagram from analog-computer
simulations of equation (5) for ~' ~ O. Figure 3.
ds-0°,andh = 0.3.
utt)
u(t)
Figure 2. Phase portraits of coexisting attractors at
Q = 0.856 and Am = 0.048. Both attractors
correspond to primary resonance.
O.
u
u
U~-Q_ ~ ~
U W
U ~
U
·~"Q;
U
US
U
t. . .1.
. . .1. . .1. . . 18
UO
W
2~1.,.
W
Phase portraits and power spectra of the
attractor and the excitation~at selected
locations near the primary resonance: (a)
T-periodic solution for [2 -= 0.933 arid An
= 0.258, (b) 2T-periodic solution for Q -
0.966 and 0tn7 = 0.268, (c ) 4T~periodic
solutions for Q = 0.942 and °{m - 0.282, (d)
8T-periodic solution for Q = 0.939 and am
= 0.284, and (e) chaotic attractor for Q -
0.937 and An,- 0.285.
136
U O
1
u
up ~Ill
° ~ .,1..
w
3
U
U
U
U
U
1
U ~
U
W
~ ~ = U(t)
1 ...
w
,,, t..1. .1. .1.
· -! .1....
(O )
w
w
I. I I l 'w U(t)
LU Figure 6. Coexisting attractors near the primary
resonance for 52 = 0.768 and ~7 = 0 217.
^3 ^1~1~l
U. W
° ~ .1
W
Figure 4. Phase portraits and power spectra of the
attractor and the excitation at selected
locations near the subharn~onic
resonance: (a) primary response fo`- Q
- 2.158 and Am = 0.107, (b) subharmonic
response for (2 = 2.102 and Em = 0.1 13,
(c) subharmonic response for 52 - 1.920
and am = 0.136, (d) subharmonic response
for Q -- 1.514 and Am - 0.218, (e)
2T-periodic subharmonic response for ha
= 1.505 and Am = 0.221, and (f) chaotic
attractor in the subharmonic response for
Q- 1.480 and An, = 0.228,
0.30
0.25
0.20
o( 0.1 5-
m
'A E,
i::: 2 1l p
p ~ ~ 2
1 ~I
U(t)
( o
·~.2'..'2:jl j
lS2 \ '"I ~
0.101 ~Al 1 1 u(t)
0.05 l Ill l l Figure 7. oexisting attractors near the
/'l ~subharmonic resonance for 52 = 1.758
0.00- ~ ' ~ ~and Am - 0.046.
O.O 0.5 1.0 1.5 2.0 2.5 3.0
Q
P3
Figure 5. Bifurcation diagram from analog-computer
sirr~ulations of equation (5) for y = 0,
B. = + 6 , and h = 0.3
137
up ~
Cu
u
is flow
u
ut it
u
u
Ou
u
3
. . . 1.
W
If.. 1,
. . . 1.
U.
Figure 8. Phase portraits and power spectra of the
response and the excitation near the
primary resonance: (a) T-periodic
response fcr Q = 1.098 and an, - ().228'
(b) 2T-periodic response for Q =- 1.098
and con, = Q.231' (c) 2T-periodic response
for Q - 1.098 and Am - 0.245, (d)
appearance of broad-band frequency
content in the 2T-periodic response for Q
= 1.093 and Am = 0.248, and (e) chaotic
attractor for Q = 1 067 and Em = 0.249.
0.30
0.25
0.20
at 0.15
m
0.10
0.05
0.00
_ .
0.0 0.5 1.0 1.5 2.0 2.5 3.0
n
Figure 10. Bifurcation diagram from analog-computer
simulations of equation (5) for ~ -- 0,
Bs = - 6°, and h = 0.3.
138
~ ~1
O . w
O
_
w
3
u33 f 1 1 1
...... 1 .
UD
cage
m~
u
Ha ~l.-l.l I ,.ll ~
tD . W
o 1
..... .... .
w
Figure 9. Phase portrait and power spectra of the
response and the excitation near the
subharmonic resonance: (a) primary
response for 52 = 2.319 and am = 0.058,
(b) subharmonic response for Q -- 1.997
and am ~~ 0.078, (c) subharmonic response
for Q = 1.794 and ten' - 0.097, (d)
2T-periodic attractor of the subharmonic
response for 52 = 1.780 and a,n = 0.099,
(e) 4T-periodic attractor of the
subharmonic response for Q = 1.763 and
Em = 0 100.
u
u
c)
u
to
u
_L 1
3
-
c~
co
°
ll ........
.1.., . .1
..1.,..1,,,l,,
0 ~
· · I.. l. · · · J I '
Figure 11. Phase portraits of the attractors near the
primary resonance and the power spectra
of the response and excitation for Q =
0 702 and Am = 0.063 and Q - 0.827 and
Em 0 057.
U(t)
OF
U(t)
Figure 12. Coexisting attractors near
subharmonic resonance for S2 =
and Am = 0.061.
Table 1. Coefficients of ship model considered.
I K ·~>0 ~ P3 2 2 [
tO0 (~0
mK2 110.mm 5.278 0.086 0.108 - 1.402 0.271 0.25/
the
1 .698
139
DISCUSSION
Alberto Francescutto
University of Trieste, Italy
First, I would congratulate the authors for this very interesting paper
indicating the possibility of new terrible possibilities in the already
complicated behaviour of a ship in longitudinal or quartering seas.
The author, in his presentation, mentions the possibility of jumps,
involving the oscillation amplitude, taking place sharply in time. In
my experience, this sharpness indicates always a few cycles of the
oscillation. Would the author comment on this point? Finally, it
seems that the consideration of the angle between the wave train and
ship's heading has been neglected in the construction of the equation
of motion (5). This, in fact, would include an explicit external
excitation also in the limiting case of pure longitudinal sea.
AUTHORS' REPLY
Dr. Francescutto's first question has to do with the time it takes an
unstable solution to grow from a given initial position to the point of
capsize. The process may involve a fraction of a cycle to many
cycles of oscillation, depending on the initial conditions and the
scenario through which the vessel capsizes. In the case of capsize
through saddle-node bifurcations, the capsize time may be very short.
On the other hand, in the case of capsize through period-doubling
bifurcations and chaos, the capsize time may be very long. In
reference to the consideration of the angle between the wavetrain and
the ship's longitudinal axis, we agree that it is not explicitly shown
in equation (5). However, the governing equation is general and can
treat arbitrary orientations by properly adjusting the excitation
parameters in equation (10).
DISCUSSION
Hongbo Xu
Massachusetts Institute of Technology, USA (China)
You have shown that the nonlinear analysis is a very useful tool of
studying the ship motion. When this one-degree freedom system
(rolling of a ship in quartering seas) bifurcates, the magnitude of the
ship motion undergoes a jump. There must be energy exchanges
between the ship and waves or energy transfer between the potential
component and kinetic component. I think it is important to identify
the energy-sharing mechanism in order to understand the dynamics of
the system. Perhaps you also have this kind of results to show us?
AUTHORS' REPLY
Dr. Xu's question addresses the energy transfer mechanism between
the wave and boa, as well as the exchange between the potential and
kinetic energies. Understanding of these fundamental mechanisms is
of primary importance as Dr. Xu points out. The basic mechanism
is the adjustment of the phasing between the vessel motion and the
wave.
DISCUSSION
Hang S. Choi
Seoul National University, Korea
The equation you have used seems very complicated; cuffing
oscillator for righting moment and Van der Pot oscillator for
damping. In addition to these, the biased initial position is
introduced as a control parameter. However, the resulting motion of
this complex system depends strongly on the initial condition of the
motion. Would you comment how the initial condition can be
determined in your mathematical model?
AUTHORS' REPI~Y
Dr. Choi's question addresses an important aspect of the mathematical
model. The solution to the differential equation governing the
oscillation of the vessel depends on the initial conditions in the form
of position and velocity. Furthermore, there might be many possible
solutions to the differential equation, depending on the initial
conditions. This is exactly the essence of the basin of attraction
shown in the paper. Each set of initial conditions, as an independent
variable, is located on the phase plane and the solution reached from
that point is determined. Therefore, the motion is strongly dependent
on the initial conditions.
In summary, the initial conditions are independent variables which
must be specified for the particular situation. To assess the stability
of the vessel, we have to determine the solution for any physically
meaningful set of initial conditions in phase space. This generates the
domain of attraction of all possible solutions. From this information,
we need to determine which solutions pose risk and use the analysis
to evaluate the seaworthiness of the vessel under consideration.
140
