ARCHIEF
AbstractThe paper discusses the various effects of the weight distribution on the roll motion of an LNG carrier based upon model tests results. The tests consisted of extinction tests in calm water as well as roll response tests in regular beam waves.
The results of the wave tests were compared with those calculated with the much used
simplified equation for pure rolling.
A, B, C, D, E a, b, d, e A
NOTATION
coefficients of the equations of motions for roll and sway added moment of inertia
AG centre of gravity from aft perpendicular
B ship's beam
damping coefficient
C restoring coefficient pg 7 GM
CB block coefficient
CM midship section coefficient
C waterplane coefficient
transverse metacentric height
g acceleration due to gravity
polar mass moment of inertia in air with respect to the longitudinal axis through the centre of gravity = k2 PV
KM height of metacentre above base
K,, wave exciting rolling moment around centre of gravity
k wave number = g
k transverse or roll radius of gyration in air added roll radius of gyration
longitudinal or pitch radius of gyration in air kq, roll radius of gyration in water
ship's length between perpendiculars
T ship's draught
Tq, natural roll period
t time
Y,, lateral wave exciting force
y sway motion
Ya sway amplitude
a,, wave slope at water surface
u,,, wave slope amplitude at water surface = k(,.
phase angle of sway motion with respect to wave elevation at centre of gravity phase angle of roll motion with respect to wave elevation at centre of gravity instantaneous wave elevation
wave amplitude
A tuning factor =
dimensionless damping coefficient
p mass density of water
V displacement volume
TAN SENG GIE and H. L. MJNKENBERG
p
Netherlands Ship Model Basin, Wageningen, The Netherlands
Technische Hogescnoo
jF
TFE N.S.MS DeIfLOcean Engng. Vol. 4, pp. 1-13. Pergamon Press 1977. Printed in Great Britain
/
2. ¿ree
/
INFLUENCE OF WEIGHT DISTRIBUTION ON
ROLLING OF AN LNG CARRIER
: t
2 TAN SENOGir andH. L. MINKENBERO
roll angle
decrement of roll amplitude per oscillation = p,
-mean of two successive roll amplitudes
O) circular wave frequency
natural roll frequency
1. INTRODUCTION
NATURAL gas which in the past hardly fulfilled a role in the supply of energy, is nowadays for Japan, Western Europe and the United States one of the most important energy sources. While the greatest part of the gas for Western Europe is transported through pipe lines,
most of the gas for Japan and the United States is supplied from overseas by means of
specially constructed Liquefied Natural Gas tankers. For the construction of LNG carriers
and more specially for the cargo tanks, many classification societies developed structural requirements in which roll, pitch and heave values or vertical accelerations are specified Du Barry, et. al. 1971). For the determination of the tank scantlings the roll motion is
important because of its large amplitude and its relatively large induced acceleration
forces.
Due to a relatively small damping, roll motion response in regular waves is a typical
resonance phenomenon with a pronounced peak and a narrow region at the natural
frequency. Hence, the magnitude of roll in irregular seas depends to a great extent on the
roll response at resonance.
To evaluate the effect of the weight distribution on the roll response in the resonance
region, systematic tests were performed with a model of an LNG carrier lying hove-to in a
series of regular beam waves. The metacentric height was varied between 2 and 17 % of
the ship's beam, whereas for each metacentric height the transverse radius of gyration was varied between 30 and 40% of the beam. The model was fitted with bilge keels. However,
for one metacentric height of 7% of the beam, tests were repeated without bilge keels.
2. GENERAL CONSIDERATIONS ABOUT ROLLING OF SHIP iN BEAM SEAS The ship motions in regular waves are usually described by six linear coupled differential
equations of the second order. Because of the linearization of the hydrodynamic problem and of the lateral symmetry of the ship with respect to the vertical centre plane, these six equations are reduced to two sets of independent equations, viz, one set for the coupled
surge, heave and pitch motions and another set for the coupled sway, roll and yaw motions.
For a ship having a small asymmetry between the fore and aft body with respect to the lateral plane through the centre of gravity the influence of yaw on roll and sway is small in
beam seas and may be disregarded (Tasai, 1965).
The coupled equations of sway and roll (see Fig. I) are written as:
sway: (p V + a»' + bj + d + ec =
(1)
R f):
roll:
G))
, K -'Iy
Y
zFio. 1. Right-handed co-ordinate system through the centre of gravity of the ship.
without the presence of incoming waves, whereas those on the right-hand side are induced by waves with the ship restrained.
In regular waves
=
cos ot, the wave exciting lateral force and rolling moment are harmonic at a frequency equal to the wave frequency.The sway and roll motions experienced by the ship can then be described by:
Y = Ya cos(it + e)
(2)
P =
cos (wt + c)
in which Ya and Pa are sway and roll amplitudes where as and c represent the phase
angles between sway and wave and between roll and wave, respectively; the reference wave motion is considered in the origin of the co-ordinate system.
The sway and roll amplitudes and phase angles follow from equation (I), for which
the coefficients A to E and a to e and the wave exciting forces and moments can be obtained by applying the strip theory, as was done by Tasai (1965) and Vugts (1968). For this method
the added mass, damping and coupling coefficients as well as the wave exciting forces at each cross-section have to be known which can be computed by potential theory.
From equation (I) the uncoupled equations of motions are obtained by putting the coupling coefficients D = E = d = e = O; the uncoupled roll equation of motion then
becomes:
(I
+ A) ± Bcp + Cp =
(3)This equation is in principle the same as that of the much used simplified equation for
pure rolling. The difference is that the wave exciting moment contains diffraction terms due
to bodywave interactions.
In the simplified equation the exciting moment is calculated from the potential of the incident waves assuming that the waves are not disturbed by the presence of the body (FroudeKriloff hypothesis). In previous work Vugts (1968; 1970) showed that in many cases large differences could be found between the exact and the approximated exciting
moments. Model experiments confirmed that for a correct prediction of wave exiting forces and moments the diffraction terms may not be ignored (Vugts, 1970; 1968).
4 TAN SENG GIF and H. L. MJNKENBERG
Satiorì OAR
kT
(I
+ A) + Bc + Ccp = C2 e -
(4)This equation is much used since the roll motion can easily be solved once the coefficients
A and B are knownI and C = pg V ? are determined from the weight distribution.
The coefficients A and B can be evaluated from the roll extinction curve; in many cases,
however, empirical data are used.
The equation (4) sometimes predicts the roll angle reasonably well, in many cases it yields erroneous results: Tasai (1965) and Vugts (1968). Partly this is caused by ignoring the diffraction terms in the calculation of the wave exciting forces and moments, partly
because the coupling of sway into roll is disregarded. Furthermore, it should be mentioned that the damping coefficient B should take into account the important viscous contributions.
Both Tasai (1965; 1972) and Vugts (1968) used the damping coefficient derived from the roll extinction curve and obtained a satisfactorily good agreement between the calculated and experimentally determined roll motions. More experimental evidence, however, is
necessary especially since the influence of the vicsous contribution in the damping coupling terms (Vugts, 1968) and wave exciting moment (Vugts, 1970) is unknown.
3. INVESTIGATION OF ROLLiNG OF AN LNG CARRIER
In the present research the roll motion of a single screw LNG carrier with a capacity
of approximately 125,000 cubic metres (see Table I and Fig. 2) was investigated by means of experiments with a 1 : 70 scaled model.
At the draught of 11.5 m the influence of the variation of the height of the centre of
3-IO
Fio. 2. Body plan, stem and stern outlines of LNG carrier.
gravity and the transverse moment of inertia was evaluated. When keeping the draught constant, the variation of the height of the centre of gravity is equal to the variation of the
metacentric height. The investigated load variations were selected on base of data of
comparable ships for which model tests have been conducted in the Seakeeping Laboratory (Van Lammeren & Vossers, 1957) of the N.S.M.13.
It was noticed, that for LNG carriers a wide range of metacentric heights, varying from
2 to l7% of the beam, existed, whereas the transverse or roll radius of gyration in air
varied mostly from 30 to 40 % of the beam. The lower values of the metacentric height were generally found at full load, whereas in ballast conditions metacentric heights up to 17 % of the beam were noticed. The differences in roll radius of gyration depended strongly upon the type of cargo tanks installed.
The present model experiments covered above-mentioned ranges of the metacentric height and roll radius of gyrationas shown in Table 2.
TABLE 2. LOAD VARIATIONS INVESTIGATE!)
Metacentric height Roll radius of gyration
(GM) (ku)
For these load variations it was derived from the calculations of Tamura (1963) and
Tasai (1965) that for a rectangular cylinder with a midship section coefficient of 1.0 the roll
potential damping will be very small in the beamdraught ratio range of 3.1-4.1.
From experiments with two-dimensional cylinders conducted by Vugts (1968) it can also be concluded that for the LNG carrier with a beamdraught ratio of 3.65 damping
will be very small in spite of the contribution of important viscous effects in hull damping. 1-lence, it was expected that with this small own hull damping large roll motions would be experienced if bilge keels were not fitted.
Therefore, it was decided to conduct the experiments with the model provided with
TABLE I. MAIN PARTICIJLARS or LNG CARRIER
Denomination Symbol Unit Magnitude
Length between perpendiculars m 273.00
Beam M m 42.00
Draught,evenkeel T m 11.50
Displacement volume
9
m3 98,740Waterplane coefficient
Block coefficient C,CB 0.8050.749
Midship section coefficient CM
-
0.991Centre of gravity from aft perpendicular AG m 138.66
Metacentre above base KM m 17.70
Longitudinal radius of gyration in air ¡ç,, %L 24
0.02 B 0.30 B 0.32 B 0.35 B 0.07 B 0.31 B 0.35 B 0.40 R 0.092 B 0.31 B 0.37 B 0.12 B 0.30 B 0.35 B 0.39 B 0.17 B 0.30 B 0.35 B 0.40 B
6 TAN SENO GIE and H. L. MINKENBERO
bilge keels having a length of 35% of the ship length, Station 6 to i 3, and aheight of 1.4 %
of the beam. The position is given in Fig. 2. For the metacentricheight of 7% of the beam comparative tests were performed without bilge keels.
For each loading condition, a roll extinction curve was determined in calm water to
establish the natural roll period and roll damping. Based upon these results a set of regular waves was selected having periods covering the resonance region limited between 0.8 and 1.2
times the natural period. For each metacentric height the wave height was kept constant, viz. 6.1, 6.1, 4.2, 3.2 and 2.3 m for the metacentric height of 0.02, 0.07, 0.092, 0.12 and
0.17 B, respectively.
To avoid that for the wide range of natural roll periods, wave conditions had to be simulated with unrealistic waveheightwavelength ratios, the wave heights were selected
such that at roll resonance the mean wave slope at the water surface was equal, viz. 2 deg. except for the metacentric height of 0.02 B. At this loading condition the mean wave slope
was 0.6 deg., since higher waves could not be generated for such long waves due to the
limited capacity of the wave generator.
4. DESCRIPTION OF TEST PROCEDURES AND ANALYSISOF RESULTS The experiments were conducted with a wooden model having a length of approximately
4 m with which extensive resistance and propuistion tests were conducted in calm water
(Jonk & Van de Beek, to be published) while the behaviour and performance were studied in regular and irregular seas (Van Sluijs and Dommershuijzen, to be published).
For one loading condition the weight distribution in the model was adjusted on a low-mass trimming table, by means of which the exact position of the centre of gravity in the vertical and horizontal direction was obtained. The roll radius of gyration was adjusted by means of a bifilar suspension. The other loading conditions were adjusted by shifting ballast weights in fore and aft body vertically or transversely to obtainthe required
meta-centric height or transverse moment of inertia. Finally the metameta-centric height was checked in calm water by a heeling experiment.
Prior to the tests in regular waves roll extinction tests were conducted in calm water for each loading condition. The model was heeled to an angleof approximately 15 deg.
and then abruptly released. The resulting oscillations were recorded. An example of such
a roll extinction curve is shown in Fig. 3.
From this curve the natural roll period T and the dampingcoefficient were evaluated.
The natural roll period follows also from the uncoupled equation of motion (4) by putting the right-hand side of the equation equal to zero and ignoring the damping term:
T = 2it
+ A).
(5)In this equation I, represents the polar moment of inertia in air with respect to the longitudinal axis through the centre of gravity and equals p V.
The coefficient A represents the added moment of inertia, whereas C is the restoring coefficient which equals pg V GM. Defining the virtual moment of inertia as (! + A), which equals k(2 p V, equation (5) can be rewritten in the morefamiliar form:
Fío. 3. Example of a roll extinction curve. 2it k
T=
i/(g GM)
where represents the roll radius of gyration in water.
By putting the measured roll period in equation (6), for each weight distribution the
quantities kq, and = /(ke2 - k2) were calculated. In Fig. 4 the added roll radius
of gyration is given on base of the natural roll frequency.
The damping coefficient is usually expressed in the following dimensionless form:
B
=
[( + A) . C]
In this equation B represents the damping coefficient. As shown by Vossers (1959 to
0.20
ojo
Added radius of gyration
I 0 L
000
(6) (7) (8) 0 025 0,50 0.75ox, rad sec
FIG. 4. Added roll radius of gyration for different natural roll frequencies as a consequence of load variation.
1962) for the light damped roll motion the dimensionless damping coefficient v can be
determined from the extinction curves using the following formula (see Fig. 3):
Pa
8 TAN SENOGia and H. L.MININBERG
where
cp = decrement of roll amplitude per oscillation Pa = mean of two successive roll amplitudes
Zero
= cpn - Pn+i
ÇO, + Pn+i
2
Time
Fic. 6. Example of roll and wave traces.
AÂA1
vv-y
A
Roll Wave GM 0028 GM=0.07B 0 30B 0.318 0358 k =032B -0.20 k =0358I,
0.408/
/ 0.15///
1/ ./
1/ /
Withbilge 0.10 - 7,1/ keels 0.05 Without bilge keels o 5 to 15 20 5 I0 5 20 0.20 0.15 010 005 o GM= 0.12 B GM= O 78 =0308 A 0.30B 0.35B O 40B / =0.35B k=O358 - k I 15 20 0 5 IO IS 20 deg degFor each constant metacentric height a set of curves of non-dimensional damping
coefficients is given in Fig. 5 for the various roll radii of gyration.
After each extinction experiment in calm water the model was subjected to the action
of a series of regular beam waves. During the investigations the model was completely free
in its motions. It was equipped with a gyroscope to measure roll angles; simultaneously the wave was recorded by means of a wave probe. Note that all waves were calibrated at the test location before the tests with the model removed. In Fig. 6 an example of the
recordings of the roll and wave is shown.
The stationary part of the recording, being free of the transient phenomena and reflection
w rod sec 7=0.I2 B o A 0.30 B = 0.35 B ko 0.39 B 0I 0.2
FIG. 7. Roll response in beam waves for various weight distributions.
effects, was analyzed by averaging the recording over a number of cycles. The roll motion
was non-dimensionalized by dividing its amplitude ,, by the maximum wave slope k,, and is presented in Fig. 7 on base of the circular wave frequency.
GM = 0.07 B
Without bilge keels
o k,, = 0 31 B ko 0.35 B k' 04DB 0.3 GM=0 17B o = 0 30 B = 0.358 ko 0.40 B 0.4 os ç, ç, 24 20 16 2 4 24 20 16 2 GM=002 B i /ç=O30B k= 0328 = 0.35 B 04 03 CM' 0.092 B o = 0.31 B k, 0.378 06 o 02 03 04 05 03 04 0,5 06 07 03 04 05 06 07
lo TAN SENO GIEand H. L. MINKENREEG
= ./({C
- (J + A) w2}2 + B2Oi2)When the equation (5) and (7) are substituted in equation (li) the roll response can
be rewritten as:
in this figure the natura! roll frequency for each loading condition is indicated by arrows.
5. COMPARISON OF APPROXIMATED CALCULATED RESULTS AND
MODEL TEST DATA
Fundamentally, the roll motion is coupled to the sway and yaw motions. For a ship having a small asymmetry between the fore and aft body with respect to the lateral plane through the centre of gravity the influence of yaw on roll and sway in beam seas is small and may be disregarded (Tasai, 1965). Section 2 deals with the coupled equations of motions for sway and roll in beam waves at zero ship speed.
To solve these equations of motions the hydrodynamic coefficients such as added mass,
added mass moment of inertia, damping and coupling coefficients and the wave exciting
forces and moments must be known.
Many of these quantities can be computed from the potential theory, but for some of them, especially the energy dissipating terms, viscous effects can not be ignored. For roll
these effects should be taken into account to obtain reliable data at resonance (Vugts, 1968 and Tasai, 1972).
Often, however, the simplified uncoupled equation of motion for pure rolling is applied,
in particular when the roll reduction owing to the application of anti-rolling devices is
predicted, see e.g. (Bootsma & Van den Bosch, 1967; Field & Martin, 1975).
The advantage and disadvantage of this equation of motion are briefly discussed in
Section 2.
In this paper it is investigated whether this simplified equation of motion yields an
accurate prediction of the roll motion at resonance, as to the magnitude and the trend with
the variation of the loading condition.
The simplified equation of roll motion is:
kT
(4)
In equation (4) a represents the wave slope at the water surface and is defined as:
=
cos (Oit + (9)The solution of equation (5) can be written as:
P Pa COS (Oit + (10)
Substitution of the equations (9) and (10) in equation (4) gives the roll response Pal wa as a function of the circular wave frequency:
kT
-wa
/[(1 - A2)2 + vA]
(J)where A = - is the tuning factor.
(O
For roll resonance, viz. A = I, equation (12) becomes:
Pa exp (- ) exp 2k2
wa V
f.T)
With equation (13) the roll response at resonance was predicted using the damping coefficient derived from the extinction curves; the influence of roll angle on the damping
coefficient was taken into account.
The results are given in Table 3 in comparison with the measured values.
TABLE 3. COMPARISON BETWEEN CALCuLATED AND MEASURED ROLL RESPONSE AT RESONANCE
6. DISCUSSiON AND CONCLUSION
The added roll radius of gyration being a measure of added mass moment of inertia
was evaluated from the roll extinction tests. The results are presented on base of the natural
roll frequency in Fig. 4, showing that the added mass moment of inertia decreases with
increasing natural roll frequency. The scatter in this figure can be expected since the added
mass moment of inertia is not only dependent on the frequency but also on the position of the centre of gravity relative to the calm water surface as shown by Tasai (1965) and
Vugts (1968).
Based on the results in Fig. 4 it was established that on the average the roll radius of
Loading condition GM/B ku/B Roll response p/c (alculated Measured 0.02 0.30 9.1 7.2 0.32 11.1 8.5 0.35 13.3 21.5 0.07 0.31 7.1 5.3 0.35 7.8 6.7 0.40 8.8 9.3 0.07 0.31 13.8 10.7 Without 0.35 15.1 11.8 bilge keels 0.40 17.4 14.1 0.092 0.31 6.7 5.9 0.37 7.7 7.9 0.12 0.30 6.5 4.8 0.35 7.4 5.9 0.39 9.0 8.2 0.17 0.30 5.4 3.5 0.35 7.4 5.2 0.40 9.6 8.0 kT Pa e 2
12 TAN SENG GIEand H. L. MINKENBERG
gyration in water k equalled 1.1 times the radius of gyration in air With this relation
equation (6) was used to calculate the natural roll period which was in good agreement with the measured value. Apparently, for the LNG carrier investigated the natural roll period can be predicted by applying k,r, = 1.1 k. In this respect it is noticed that although due to the bilge keels the added mass moment of inertia increases, the natural roll period
becomes only slightly larger.
In Fig. 5 the dimensionless damping coefficients are given as obtained from the roll
extinction curves using equation (8) showing that the damping coefficient decreases when
the roll radius of gyration is increased. A large decrease was established for the weight distribution with a metacentric height of 2 and 17% of the beam.
In case the roll radius of gyration is kept constant the dimensionless damping coefficient appears to fluctuate with the metacentric height.
For all loading conditions investigated the damping coefficient v increases linearly with
the roll amplitude, at least up to 15 degrees. The increment is in general largest for the metacentric height of 2% of the beam.
From the comparative tests with and without the bilge keels it follows that also the
increase of the damping coefficient, due to the application of the bilge keels, becomes larger at increasing roll angles. At small roll angles the damping is about 100-200% and at larger roll angles about 200-300% larger.
The influence of bilge keels upon the roll motion in regular beam waves is shown in Fig. 7; by applying bilge keels the roll response in the resonance region is reduced to
approximately 40 0/e. Outside this region the effect of the bilge keels is negligibly small.
The roll response curves for the other loading conditions (with bilge keels) are also given ¡n Fig. 7, from which it may be concluded that:
at a constant radius of gyration the roll response at resonance is smaller when the
metacentric height is increased.
at a constant metacentric height the roll response increases with the radius of gyration.
the roll region becomes wider when the metacentric height increases.
The first trend was also found by Baitis and Wermter (1972) for a low-block and a
high-block vessel.
As to the resonance region it should be remaked that if the response curves are plotted on a non-dimensional base e.g. on base of the tuning factor A = o/w, (Vugts, 1968) or
wave lengthship length ratio (Baitis and Wermter, 1972) it will be seen that the resonance region tends to increase when the metacentric height is reduced. This trend is in agreement with that established by Vugts (1968) and Baitis and Wermter (1972).
In predicting the roll response with the simple equation of motion as outlined in Section 5
the damping coefficient is usually kept constant. From equation (13) it then follows that: increasing the metacentric height leads to a reduction of the roll response at resonance. increasing the roll radius of gyration results into an increase of the response.
As earlier discussed the damping coefficient v, becomes smaller at larger roll radius of
gyration. Consequently, the last mentioned trend of' the roll response remains valid.
However, since the damping coefficient does not have a unique relationship with the metacentric height, the trend of the roll response cannot be judged from equation (13)
beforehand without performing the calculations as done in Section 5 taking into account
the experimentally determined damping coefficient.
response at resonance can be fairly well predicted by the simple theory; however, the magnitude of the response is mostly over-estimated. This is not in agreement with the
findings ofBaitis and Wermter (1972); they found that the uncoupled equation of motion predicted lower roll response at resonance. Tasai (1965) shows that the usefulness ofthe
uncoupled equation of motion cannot be evaluated beforehand.
As shown by Tasai (1965) and Vugts (1965) at a constant draught the coupling terms
depends strongly on the weight distribution. For two-dimensional bodies Vugts (1968)
concluded that the general tendencyofthe coupling terms with swaying is to lower and to narrow the roll response, especially at the high frequency side.
Summarizing it can be concluded that for the LNG carrier considered the simple theory
predicts fairly well the tendency of the roll response at resonance but leads to an
over-estimationofthe magnitude of the response. Taking into account the coupling effects of
swaying motion and using the damping coefficient experimentally determined a better
prediction of the roll response may be expected. REFERENCES
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