ON THE USE OF U-TYPE STABILIZING TANKS FOR REDUCING ROLL
MOTIONS IN HEAD SEAS
Marcelo A. S. Neves, LabOceano - COPPE/UFRJ
Jorge A. Merino, COPPE/UFRJ
Claudio A. Rodríguez, LabOceano - COPPE/UFRJ
Luiz Felipe N. Soares - UFRJ
SUMMARY
Many investigations have been conducted on the use of U tanks in the control of the roll motions in beam seas. However,
very few papers have dealt with their application in the case of parametric resonance in longitudinal waves. The present
paper describes the development of a mathematical model in which the fluid motion inside a U tank is non-linearly
coupled to the heave, roll and pitch motions of the ship.
A transom stern small vessel, a vessel well known to be quite prone to parametric amplification is employed in the study.
The results are presented in the form of limits of stability, with encounter frequency and wave amplitudes as parameters.
Distinct dynamical characteristics are discussed and conclusions are drawn on the relevant parameters for the efficient
control of the roll amplifications in head seas.
system CXYZ moving at the average ship speed (U),
such that at instant t=0 the plane XY coincides with the
calm water free surface with point C at the same
vertical of the centre of gravity G of the ship. A second
coordinate system is the moving frame Oxyz . This
second system is fixed to the ship such that the plane
xy coincides initially with the ship waterplane in calm
1. INTRODUCTION
In longitudinal waves a ship may be forced to roll
heavily. This is not due to direct wave excitation.
Instead, an internal excitation is the mechanism
responsible for the so-called parametric excitation. In
face of some serious recent incidents of great roll
amplification in head seas, as reported in [1], great
attention has been given recently to head seas
parametric resonance, see for example refs. [3, 6, 11],
and possible control of head seas parametric rolling.
water and axis Ox belongs to the diametral plane,
being positive in the sense of ship speed, axis Oy
points to portside and Oz axis passes always by the
vertical line containing the ship centre of gravity G,
positive upward. The two reference frames are shown in
Figure 1.
Refs. [2, 4, 10, 12, 14, 15] have discussed in detail the
application of anti-rolling devices and particularly antirolling tanks (ART) to the reduction of roll motions of
ships subjected to external wave excitations.
The present paper summarises the development of a
non-linear mathematical model of ART dynamics
derived in [5] compatible with the 3rd order
mathematical model introduced by Neves and
Rodríguez [7, 8] for the analysis of parametric rolling in
head seas. In the present model the fluid motion inside a
U tank is non-linearly coupled to the heave, roll and
pitch motions of the ship. A transom stern small vessel,
a vessel well known to be quite prone to parametric
amplification is employed in the study. The results are
presented in the form of limits of stability, with
encounter frequency and wave amplitudes as
parameters. Distinct dynamical characteristics are
discussed and conclusions are drawn on the relevant
parameters for the efficient control of the roll
amplifications in head seas.
Figure 1 – Inertial and fixed coordinate axes
Geometric characteristics of the U tank are introduced
in Figure 2. Two vertical reservoirs are connected by a
transversal duct, with all elements having rectangular
constant cross sections. At the centre of the duct a pump
is contemplated as a possible device to control the fluid
motion. Fluid motion, assumed to be unidirectional, is
described by fluid displacement Z (t ) and relative fluid
2. COORDINATE SYSTEMS AND TANK
DEFINITIONS
In order to describe the ship motions and internal fluid
motion two reference axes are employed. An inertial
Session A
velocity Z& (t ) . It should be noted that point O not
51
~
necessarily coincides with G, the ship centre of gravity,
assumed to be positioned (unmovable) at (0,0, z G ) . In
later
developments
angular
chosen origin. A is also a 6x6 matrix, whose elements
represent hydrodynamic generalized added masses.
~
B (φ& ) describes the coefficients of the hydrodynamic
reactions dependent on ship velocities (damping), and
may incorporate non-linear terms in the roll equation.
variable
⎛ Z (t ) ⎞
⎟⎟ will be employed instead of the
⎝ BW ⎠
τ (t ) = tan −1 ⎜⎜
r r
C r ( s , ζ) is a 6x1 vector which describes non-linear
linear variable Z(t). Its geometric interpretation is made
clear by noting Figure 3.
restoring forces and moments dependent on the relative
motions between ship hull and wave elevation ζ(t). On
the right hand side of equation (1), the generalized
r
represents wave external
vector C ext (ζ, ζ& , &ζ&)
excitation, usually referred to in the literature as the
Froude-Krilov plus diffraction wave forcing terms,
dependent on wave heading χ , encounter frequency ωe,
wave amplitude Aw and time t. Finally, generalized
vector
r
r r r
C t ( Z , Z& , Z&&, s , s& , &s&) represents forces and
moments acting on the ship due to the fluid motion
inside the tank. For a ship without tank, equation (1)
reproduces the system of non-linear equations
introduced in Refs. [7, 8, 9].
4. DERIVATION OF THE ART EFFECTS
Under the assumption of unidirectional fluid motion,
the differential momentum equations of internal fluid
motion may be somewhat simplified by considering the
dynamics of elementary fluid volumes. Figure 4
illustrates the free body diagram of an elementary fluid
volume considered at the portside reservoir. Similar free
body diagrams are defined for the left and right
elements of the duct and starboard reservoir. At each
element, the momentum rate is defined as: ρ t dva A
Figure 2 – Schematic representation of U tank
where dv is the elementary volume. Balance of
momentum takes into account the external effects of
weight, surface tension on the wall and pressure
variation, being expressed as:
dF = dFw + dFwall + dFp = ρt Ar a A dh
Figure 3 – Alternative angular variable
where
ρt
is the internal fluid density,
(2)
Ar is the cross
3. GENERAL EQUATIONS OF MOTION
sectional area of the reservoir, a A is the absolute
Non-linear equations of motion may be represented as:
acceleration of the elementary volume considered and
dh is an increment defined along the reservoir axis, as
indicated in Figure 2. It is pointed out that the mass of
an elementary volume is dmt = ρt Ar dh .
~ ~ r ~ r r r
( M + A )&s& + B (φ&) s& + C r ( s , ζ ) =
r
r
r r r
C ext (ζ , ζ& , ζ&&) + C t ( Z , Z& , Z&&, s , s& , &s&)
(1)
5. DERIVATION OF FLUID ACTIONS ON THE
v
where vector s (t ) represents rigid body motions in six
PORTSIDE RESERVOIR
~
degrees of freedom. In equation (1) M is a 6x6 matrix
which describes hull inertia characteristics. Its elements
are: m, the ship mass, first moment mz G , Jxx, Jyy and
In the following, expressions for all the terms in
equation (2) will be detailed for the portside reservoir.
a) The elementary weight is defined as:
Jzz the mass moments of inertia in the roll, pitch and
yaw modes, respectively, and J xz , the roll-yaw product
of inertia, all moments taken with reference to the
Session A
52
where Lx is the longitudinal position of the tank
expressed in the ship coordinate system.
dv 0
= u&i + v&j + w& k + Ω × v 0
dt
dr
v 0 = 0 = ui + vj + wk
dt
Ω = pi + qj + rk
& = p& i + q&j + r&k
Ω
a0 =
(3-1)
N x1 = ρt Ar {[− gsinθ + u& + qw − rv −
−(q 2 + r 2 ) Lx − (r& − pq) Bw + 2qZ& ]
N y1 = ρt Ar {[ gsinθ cos φ + v& + ru − pw −
b) The force on the wall, as indicated in Figure 4, is
−( p 2 + r 2 ) Bw + (r& + pq) Lx − 2 pZ& ]
expressed as:
τ r Per
+ g cos φ cos θ + w& + pv −
ρt Ar
Pp − P1 = − ρt {[
is the surface tension on the walls of the reservoir,
−qu − (q& − pr ) Lx + ( p& + qr ) Bw + Z&&]
Per = 2(Wr + Lt ) is the perimeter of the cross section
and
transversal
Equivalent derivations have been developed for the
other sections of the tank. Complete details may be
found in [5]. Integrating the forces and moments along
the whole length of the tank the non-linear equation of
fluid motion may be obtained:
B
Ped
H
Per
E
Wr ( w − r ) Z&& +
Bτ −
H τ + KZ& 2 −
ρA w d ρA r r
2 ρ A Z&
H
W
(3-3)
d) Elementary volume:
dv r = Ar dh
d
(3-4)
(3-5)
t
r
t
r
Pp − Ps
where E is the power delivered to the fluid by a pump
and K is a localized head loss. In equation (5) the
tangential tensions on the walls ( τ r ,τ d ) may be
(3-6)
(3-7)
expressed as [13]:
are the relative velocity and acceleration, respectively.
Additionally,
Session A
d
(5)
where:
rB1 = Lxi+Bw j − ( H 2 + h ) k
t
− [ gsinφ cos θ + v& + ru − pw + ( r& + pq ) Lx +
2 ρt
+ ( p& − qr ) Lz − ( p& + qr ) H r ]Bw
=
& r + 2Ω× v + a
a A = a 0 + Ω× ( Ω× rB ) + Ω×
B
B
B
B1
r
+[ g cos φ cos θ + w& + pv − qu − ( q& − pr ) Lx + ( p 2 + q 2 ) H 2 ]Z
e) Absolute acceleration:
v B1 = Z& (t )k
a = Z&&(t )k
(4-3)
1
( H r − Z ) + ( p 2 + q 2 )[ L2z − ( H 2 + Z ) 2 ]}
2
components on the walls.
c) Net pressure force is expressed as (see Figure 4):
dFp = Ar dPk
(4-2)
1
( H r − Z ) + ( p& − qr )[ L2z − ( H 2 + Z )2 ]}
2
(3-2)
where:
longitudinal
(4-1)
1
( H r − Z ) − (q& + pr )[ L2z − ( H 2 + Z ) 2 ]}
2
the unit vector along the vertical.
of the reservoir and
N x , N y are the
(3-12)
reservoir are obtained as:
+ cos φ cos θ k)dh
r
where gu = −sinθ i + sinφ cos θ j + cos φ cos θ k is
τr
(3-11)
reservoir. Subsequently integrating from h = Z to
h = H r , expressions for forces acting on the portside
r
dFw = − ρt gAr gu dh =
dFwall = dN x i+dN y j − τ r Per dhk
(3-10)
Substitution of equations (3.1 ~ 3.11) into equation (2)
will result in explicit expressions for the unknown
differentials dN x ,dN y and dP for the portside
Figure 4 – Elementary fluid volume: port side reservoir.
− ρt gAr (−sinθ i + sinφ cos θ j
(3-9)
τ r ,d = f
(3-8)
53
γ Z& 2
8g
X ta = − ρt Ar {C1[− gsinθ + u& + qw − rv −
where f is a friction factor, assumed to be dependent on
Re =
&
ZD
r ,d
2
+C2 ( p& − qr ) + 2 Bw ( p 2 + r 2 ) Z + 2 Bw Z&&}
Z ta = − ρt Ar {C1[ g cos φ cos θ + w& + pv − qu −
− L (q& − pr )] + 4 B pZ& +
what part of the tank the Reynolds number is computed.
The friction factor may be estimated by means of semiempirical formulas found in the literature, [13].
x
K ta = − ρt Ar {2 Bw ( H r + Lz ) Z&& + 4 H 2 pZZ& +
damping actions are proportional to fluid velocity Z& ,
such that an equivalent damping ( Bττ ) will be defined
+1/ 3Bw2 (C1 + 4 H r )( p& + qr ) −
−2 Bw [ g cos φ cos θ + w& + pv − qu −
as:
(7-4)
− Lx (q& − pr ) + H 2 (q 2 − r 2 )]Z +
+2 / 3(3B% w L2z + 3H 22 H r − 3H 2 Z 2 + 3H 2 H r2 + H r3 )
( p& − qr ) + C2 [ gsinφ cos θ + v& − pw + ru +
Ped
Per
Bwτ d −
H τ + KZ& 2 ≈ Bττ Z&
ρt Ad
ρt Ar r r
+ Lx (r& + pq )]}
Internal damping coefficient Bττ may be determined
M ta = − ρt Ar {2 Lx ZZ&& + 4( H 2 qZ + Bw Lz r − Bw Lx p) Z& −
− Lx C1[ g cos φ cos θ + w& − qu + pv − Lx (q& − pr )]
(7-5)
+2 / 3(3B% L2 + 3H 2 H − 3H Z 2 + 3H H 2 + H 3 )
by means of decay tests:
Bττ
B
H
2 gWr ( w − r )
H d Wr
w
2
z
r
2
2
r
r
(q& + pr ) − 2 Bw H 2 (r& − pq) Z + 2 Bw Lx ( p& + qr ) Z −
−C2 [− gsinθ + u& − rv + qw + Lx ( p 2 − r 2 )]}
As additional simplifications it will be assumed that the
pump is not operating (E=0) and the portside and
starboard pressures are equal to the atmospheric
pressure ( Pp − Ps = 0 ), such that the tank is then in a
N ta = − ρt Ar {2 Lx Bw Z&& + 4( Lx pZ − Bw H r q ) Z& +
+2 / 3B 2 ( B% + 3H )( r& − pq) +
w
w
r
+2 Bw [− gsinθ + u& − rv + qw + Lx ( p 2 − q 2 ) −
− H 2 (q& + pr )]Z + C2 Lx ( p& − qr ) +
purely passive mode of operation.
(7-6)
2C1 Lx [ gsinφ cos θ + v& + ru − pw + Lx (r& + pq)] +
6. FORCES AND MOMENTS EXERTED BY THE
+2 Bw Lx ( p 2 + r 2 ) Z }
TANK ON THE SHIP
where:
Tank force for an elementary volume of fluid is
expressed as:
B
B% w = w , C1 = 2( H r + B% w ) ,
R
r
dFtank = − (a A +ggu )dmt
C2 = L2z − H 22 − Z 2 + 2 Lz B% w
and the corresponding moment as:
~
and T is the 3-D matrix of rotations.
r
= − rB × (a A +ggu )dmt
Thus, equation (1), with the last term on the right hand
side given by equations (7), together with equation (4)
will form a set of 7 non-linear coupled second order
differential equations describing the six degrees of
freedom ship motions and fluid motion.
r
&&, sr, sr& , &sr&) ,
such that the generalized vector C t ( Z , Z& , Z
defined in equation (1), may be separated in:
7.
PARAMETRIC
RESONANCE
LONGITUDINAL WAVES.
(6)
IN
Focusing on the problem of parametric rolling in
longitudinal waves, the dynamic coupled ship/tank
problem may be investigated with reference being made
to the heave-roll-pitch-tank problem, resulting in a four
with the following expressions:
Session A
(7-3)
w
+C2 ( p 2 + q 2 ) − 2 Bw ( p& + qr ) Z − 2ZZ&&}
As a further approximation, it will be assumed that the
⎡X a ⎤
⎡ Ka ⎤
⎡ C t1 ⎤
⎡Ct 4 ⎤
⎢C ⎥ = T~ ⎢ Y ⎥ ; ⎢C ⎥ = T~ ⎢ M ⎥
⎢ a⎥
⎢ a⎥
⎢ t2 ⎥
⎢ t5 ⎥
⎢⎣ Z a ⎥⎦
⎢⎣ N a ⎥⎦
⎢⎣C t 3 ⎥⎦
⎢⎣Ct 6 ⎥⎦
(7-2)
x
width ( Wr ) or the duct height ( H d ) depending on
dM tank
w
Yta = − ρt Ar {C1[ gsinφ cosθ + v& + ru −
− pw + L (r& + pq )] + 4 pZZ& +
ν
The characteristic length D r , d shall be the reservoir
ηt =
(7-1)
− Lx (q 2 + r 2 )] + 2 Bw (r& − pq ) Z −
−C (q& + pr ) − 4( B r + qZ ) Z& }
the Reynolds number:
54
investigate the general trends of the influence of the
ART parameters, systematic simulations have been
performed in the case of a fishing vessel denominated
TS, described in [7]. Figure 5(upper) shows the heave,
roll and pitch time series corresponding to conditions
( GM = 0.37m , Fn = 0.30, ω e = 2.1ω n 4 ) for which
degrees of freedom problem. Neglecting surge, sway
and yaw motions ( X ta = Yta = N ta = 0 ), and
redefining the coefficients into a derivative
nomenclature introduced in Ref. [5], results in the
following four equations:
& & + Z & &φ&2 + Z & &θ& 2 +
Z ta = Zτ &&z &&
z + Zτθ&&θ&& + Zφτ& &φτ
τφφ
τθθ
&& + Z ττ&& +
Zτφθ cos φ cos θ + + Z φτ&& φτ
ττ&&
intense parametric amplification occurred for the ship
without tank. These conditions are compared in the
lower graph with situations of reduction of the roll
motion due to the tank. It is also observed that no
significant changes in the vertical motions are detected.
The complete cancellation of parametric rolling may be
seen to be true even for extreme cases like in Figure 6.
This is a condition in which the ship, without tank,
would encounter capsize. By application of a tank in
heavy condition, even not being closely tuned, the roll
motion completely disappears.
(8-1)
&2 2
&2 2
+ Zφφττ
& & φ τ + Zθθττ
&& θ τ
&& + K & z&φ& +
K ta = Kτ&&τ&& + Kτφ&&φ&& + K &&zτ &&
zτ + Kθτ&& θτ
τ z&φ
& & + K sinφ cos θ + K & & θ& 2τ +
+ Kτφθ& &φθ
τφθ
θθτ
(8-2)
&& 2
&
+ Kφττ
&& φτ + Kφττ
& &φττ& + Kφθτ cos φ cos(θ )τ +
& 2 + K & & φθτ
&& 2
&φτ
+ Kφθττ sinφ cos(θ )τ 2 + K z&φττ
& z
φθττ
A question arises as to what combinations of wave
amplitude and wave length the tank may induce
beneficial reduction or elimination of the undesirable
parametric rolling. Given the broad range of relevant
parameters involved, it becomes necessary to map the
areas of stable and unstable motions. Figure 7 shows the
results for the limits of stability corresponding to ship
TS without tank, for two speeds. It is observed that the
area of unstable motions is large in both speeds.
Clearly, the condition Fn = 0.30 corresponds to more
&& +
M ta = M τ &&z &&
z + M τθ&&θ&& + M τθ sinθ + M φτ&& φτ
& & + M & &φ&2 + M & z&θ& + M cos φ cos θ + (8-3)
+ M φτ& &φτ
τφθ
τφφ
τ z&θ
2
&&
&
+ M ττ&&ττ&& + M θττ
&& θττ + M θττ
& &θττ& + M θττ sinθτ +
& & + M & & φ&2τ 2 + M & θ& z&τ 2
+ M & & φθτ
φθτ
φφττ
θ z&ττ
&& + T sinφ cos θ +
Tτ&&τ&& + Tτ&τ& + Tφ&&φ&& + T&&zτ &&
zτ + Tθτ&& θτ
φθ
& & + T & & φ&2τ + T & & θ& 2τ +
+Tz&φ& z&φ& + Tφθ& &φθ
φφτ
θθτ
(8-4)
+Tφθτ cos φ cos(θ )τ = 0
intense amplifications, what had already been
demonstrated by Neves and Rodríguez [8]. Here we
have chosen to perform the mappings for this ship
forward speed with stronger roll amplification, in order
to reveal the whole spectrum of possible responses. In
Figures 8 and 9 the limits of stability are presented for
the ship with tank for a condition with 3% of mass of
water with respect to the ship mass, and damping levels
of 0.2 and 0.3, respectively, and different tank tunings,
ωt / ω n 4 . It may be observed that in the cases of
The new derivative coefficients are shown in Table 1 in
terms of the tank parameters. The set of four non-linear
ordinary differential equations may be numerically
solved for a given vessel and a specified tank positioned
at some defined position on the vessel. It may be
noticed that in the particular case of roll/tank linear
equations the problem simplifies to:
K ta = Kτ&&τ&& + Kτφ&&φ&& + Kτφθ φ + Kφθτ τ
T τ&& + T τ& + T τ = −(T φ + T&&φ&&)
τ&&
τ&
φθτ
φθ
ωt / ω n4 = 1.0 and ωt / ω n 4 = 1.1 better results are
obtained, in the sense that the area of unstable motions
substantially reduced and the unstable roll motions have
small amplitudes.
φ
which is analogous to the well known system of
equations derived in [4]. The tank natural frequency ωt
is:
ωt =
Tφθτ
Tτ&&
=
Figures 10 and 11 show the effects on the limits of
stability due to changes in the ratio of mass of the tank
fluid to ship mass considering the tuning ω t = ω n 4 and
g
Wr Bw
− Hr
Hd
two levels of damping ratio. It may be observed that a
ratio of mt = 2m or more is required for the unstable
areas to shrink. Unfortunately, that can only be
achieved with a price to be paid in terms of deadweight.
Residual roll amplification are encountered in the range
of higher frequencies. As the mass mt increases, the
8. RESULTS
ART main dimensions are given in Table 1. Five values
have been defined for the parameter H d in order to
obtain five distinct ratios for the tuning
ω t / ω n 4 . The
unstable area moves slightly to the right and markedly
upwards, which is the relevant effect to be expected in
the design of an efficient ART. In Figure 11 the same
mass variation is simulated, but now with a higher value
for the internal damping. It is observed that in this
tank horizontal dimension Lt was chosen to be the free
parameter to allow variations in the ratio of mass of
internal water to ship mass mt / m . In order to
Session A
55
condition more substantial reductions in the area of
unstable motions are found. Only at very high
frequencies some amplification persists.
It is further observed that no significant roll
amplification was observed inside the area of
fundamental resonance for any set of tank parameters.
Further research are expected to be conducted on this
very extensive subject, particularly involving an
experimental test program.
4.
5.
9. CONCLUSIONS
6.
Non-linear equations for the coupled motions of a ship
and the water inside a tank have been introduced. The
set of equations is defined to the third order, compatible
with the set of equations previously employed by Neves
and Rodríguez [7, 8]. In general the simulations have
indicated that it is possible to control strong parametric
resonance if the correct choice of parameters is made in
the first region of instability. Mapping of stable and
unstable regions have been obtained for different
parameters of the ART. It has been shown that tuning of
the tank and ship natural frequencies and internal mass
are the crucial parameters for the successful utilization
of an U-tank in reducing the risk of inception of
parametric resonance. According to the results, internal
damping plays a less marked influence. These are
aspects that should be examined in more detail in the
future. Numerical limits of stability may be seen to be
an efficient tool in order to get a comprehensive
assessment of the ART characteristics. Important to say,
it is shown that for the range of parameters simulated,
an ART may eliminate roll amplification at some
conditions but persists (or appears) at some others. It is
expected that model tests will be of great help in the
validation of the present numerical simulations.
7.
8.
9.
10.
11.
ACKNOWLEDGEMENTS
The present investigation is supported by CNPq within
the STAB project (Non-Linear Stability of Ships),
CAPES and LabOceano.
12.
REFERENCES
13.
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Hill, 1999.
YOUSSEF, K. S., RAGAB, S. A., NAYFEH,
A. H., MOOK, D. T., “Design of Passive AntiRoll Tanks for Roll Stabilization in the
Nonlinear Range”. Ocean Engineering, vol. 29,
no. 2, pp. 177-192, 2002.
YOUSSEF, K. S., MOOK, D. T., NAYFEH,
A. H., RAGAB, S. A., “Roll Stabilization by
Passive Anti-Roll Tanks Using an Improved
Model of the Tank-Liquid Motion”, Journal of
Vibration and Control, vol. 9, no. 7 (July), pp.
839-862, 2003.
Table 1a: Heave/tank coefficients
Zτ &&z = ρt Ar C1
Zτθ&& = − ρt Ar C1 Lx
Zττ&& = −2 ρt Ar Bw2
Zφτ&& = −2 ρt Ar Bw2
Zφτ& & = 4 ρt Ar Bw2
Zτφφ& & = ρt Ar ( L2z − H 22 + 2 Lz B% w )
Zτθθ& & = ρt Ar ( L2z − H 22 + 2 Lz B% w )
Zτφθ = ρt Ar C1 g
Table 1b: Roll/tank coefficients
Kτφ&& = 2 ρt Ar [ B% w L2z +
Kτ&& = 2 ρt Ar Bw2 ( Lz + H r )
2
Kθθτ
& & = −2 ρt Ar Bw H 2
H 22 H r + H 2 H r2 + H r3 3 +
Bw2 (C1 + 4 H r ) 6]
Kφττ& = 4 ρt Ar Bw2 H 2
Kφθτ = −2 ρt Ar Bw2 g
2
Kφττ
&& = −2 ρt Ar Bw H 2
2
K z&φττ
& = ρ t Ar Bw
Kφθττ = − ρt Ar Bw2 g
2
Kφθττ
& & = − ρt Ar Bw Lx
Table 1c: Pitch/tank coefficients
M τ &&z = − ρt Ar Lx C1
M τθ = ρt Ar g ( L2z − H 22 +
2 L B% )
z
M ττ&& = 2 ρt Ar Bw2 Lx
M τθ&& = 2 ρt Ar ( B% w L2z +
H 22 H r + H 2 H r2 + H r3 3 +
w
L2xC1 2)
M φτ&& = 2 ρt Ar Bw2 Lx
M τφφ& & = − ρt Ar ( L2z − H 22 +
M φτ& & = −4 ρt Ar Bw2 Lx
M τφθ = ρt Ar Lx C1 g
2 Lz B% w )
2
M θττ
& & = 4 ρt Ar Bw H 2
M τ z&θ& = − ρt Ar ( L2z − H 22 +
2 L B% )
M τ z&θ& = − ρt Ar ( L2z − H 22 +
2 L B% )
M φφττ
& & = ρ t Ar B Lx
M θ&z&ττ = ρt Ar Bw2
z
M φθτ
& & = 2 ρt Ar B H 2
2
w
w
z
2
w
2
M θττ
&& = −2 ρt Ar Bw H 2
w
Table 1d: Tank coefficients
⎛B
H ⎞
Tτ&& = 2 ρt Ar B Wr ⎜ w − r ⎟
⎝ H d Wr ⎠
T&&zτ = 2 ρt Ar Bw2
Tτ& = 2 ρt Ar Bw2 Bττ
Tφ&& = 2 ρt Ar Bw2 ( Lz − H r )
Tz&φ& = −2 ρt Ar Bw2
Tφθ = 2 ρt Ar Bw2 g
Tθτ&& = −2 ρt Ar Bw2 Lx
Tφθ& & = 2 ρt Ar Bw2 Lx
Tφθτ = 2 ρt Ar Bw2 g
2
Tφφτ
& & = 2 ρt Ar Bw H 2
2
Tθθτ
& & = 2 ρt Ar Bw H 2
2
w
Table 2: ART main dimensions
B w (m) Wr (m) H r (m) H d (m) ωt (rd/s) ωt / ωn 4
1
2
3
4
5
Session A
3.00
3.00
3.00
3.00
3.00
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
1.50
0.251
0.304
0.360
0.419
0.480
57
0.773
0.859
0.944
1.030
1.116
0.900
1.000
1.100
1.200
1.300
Figure 5 – Time series for ship TS, heave, roll, pitch, GM=0.37m, Fn=0.20, Aw=0.6m. Upper graph, without tank.
Lower graph, with tank ( ω t = 1.3ω n 4 , mt = 3%m , η t = 0.3 ).
ω e = 2.45ω n 4 , Aw=0.7m. Without tank, capsize condition;
tank, roll disappears ( ω t = 1.3ω n 4 , mt = 3%m , η t = 0.3 ).
Figure 6 – Roll motion for ship TS, GM=0.37m, Fn=0.30,
with
Figure 7 – Ship TS without tank. a) Fn=0.20; b) Fn=0.30.
Session A
58
Figure 8 – Ship TS, GM=0.37, Fn=0.30, with tank. Limits for different
ωt / ω n 4 for mt = 3%m and η t = 0.2 .
Figure 9 – Ship TS, GM=0.37, Fn=0.30, with tank. Limits for different
ωt / ω n 4 for mt = 3%m and η t = 0.3 .
Session A
59
Figure 10 – Ship TS, GM=0.37, Fn=0.30, with tank. Limits for different
mt for ω t = ω n 4 and η t = 0.2 .
Figure 11 – Ship TS, GM=0.37, Fn=0.30, with tank. Limits for different
mt for ω t = ω n 4 and η t = 0.3 .
Session A
60