HYDRODYNAMIC INTERACTION BETWEEN FLNG VESSEL AND LNG CARRIER IN SIDE BY SIDE CONFIGURATION

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648
 
2012,24(5):648-657 
DOI: 10.1016/S1001-6058(11)60288-6 
HYDRODYNAMIC INTERACTION BETWEEN FLNG VESSEL AND LNG 
CARRIER IN SIDE BY SIDE CONFIGURATION*
ZHAO Wen-hua, YANG Jian-min, HU Zhi-qiang 
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China, 
E-mail: zwzldh@163.com 
(Received March 14, 2012, Revised April 23, 2012) 
Abstract: The Floating Liquefied Natural Gas (FLNG) is a new type of floating platform for the exploitation of stranded offshore 
oil/gas fields. The side by side configuration for the FLNG vessel and the LNG carrier arranged in parallel is one of the possible 
choices for the LNG offloading. During the offloading operations, the multiple floating bodies would have very complex responses
due to their hydrodynamic interactions. In this study, numerical simulations of multiple floating bodies in close proximity in the side 
by side offloading configuration are carried out with the time domain coupled analysis code SIMO. Hydrodynamic interactions 
between the floating bodies and the mechanical coupling effects between the floating bodies and their connection systems are 
included in the coupled analysis model. To clarify the hydrodynamic effects of the two vessels, numerical simulations under the same 
environmental condition are also conducted without considering the hydrodynamic interactions, for comparison. It is shown that the 
hydrodynamic interactions play an important role in the low frequency motion responses of the two vessels, but have little effect on 
the wave frequency motion responses. In addition, the comparison results also show that the hydrodynamic interactions can affect the 
loads on the connection systems. 
Key words: Floating Liquefied Natural Gas (FLNG), hydrodynamic interactions, side by side operation, relative motions 
Introduction�
Due to the high oil prices and the steep increase 
in the natural gas demand triggered by Japanese 
nuclear disaster, the stranded offshore gas fields, 
which were considered not quite favorable projects in 
the past, are becoming more and more attractive. To 
exploit those offshore gas fields effectively and eco- 
nomically, the Floating Liquefied Natural Gas 
(FLNG), as a new concept of offshore unit, was pro- 
posed recently. The configurations of an FLNG vessel 
and an LNG carrier in side by side or tandem arrange- 
ments are two possible ways of the LNG offloading 
operation. In the case of the side by side configuration, 
the FLNG vessel and the LNG carrier are moored in 
 
* Project supported by the China National Scientific and 
Technology Major Project (Grant No. 2011ZX05026-006-05), 
the Science Foundation of Science and Technology Commi- 
ssion of Shanghai Municipality (Grant No. 11ZR1417800). 
Biography: ZHAO Wen-hua (1986-), Male, Ph. D. Candidate 
Corresponding author: YANG Jian-min, 
E-mail: jmyang@sjtu.edu.cn 
close proximity. Therefore, the hydrodynamic intera- 
ctions between the two vessels can not be ignored. 
The hydrodynamic interactions might significantly 
affect the motion responses of the vessels and make 
the relative motions of the multiple bodies very com- 
plex. Furthermore, the investigation of the hydrodyna- 
mic interactions also helps in the prediction of the 
collision events between the two vessels. 
The hydrodynamic interactions of multiple floa- 
ting vessels in the parallel configuration were much 
studied. In the early studies of multiple bodies, the 
dynamic cross-coupling among the motions of the 
bodies and the hydrodynamic interactions between the 
multiple bodies were not considered. Later, a theory 
on the hydrodynamic interactions of parallel floating 
bodies based on a two-dimensional diffraction theory, 
was proposed. However, an unphysical phenomenon 
exists in the numerical simulations based on the pote- 
ntial theory due to the fluid resonant response in the 
gap of the multiple bodies. Buchner et al.[1] introduced 
a rigid lid on the free surface within the multi-body 
diffraction analysis to suppress the unrealistic reso- 
nant wave oscillations, which is an improvement over 
649
the previous methods. Using the same measure to deal 
with the resonant wave oscillations, Naciri[2] carried 
out extensive time domain analyses of the side-by-side 
moored vessels and verified the simulation results 
through model tests. Pauw et al.[3] considered two 
ships in a side by side arrangement and replaced the 
rigid lid with a numerical damping lid. They sugge- 
sted that the damping parameter should be adjusted in 
such a way that the drift forces rather than the linear 
quantities are optimal with respect to the model tests. 
In order to avoid the restriction that each interacting 
body must be far enough apart from the other bodies, 
Kashiwagi and Shi[4] solved the integral equation of 
the diffraction potential by the Higher-Oder Boundary 
Element Method (HOBEM). They found that the sma- 
ller the separation distance between bodies, the larger 
deviation of the pressure distribution obtained based 
on the wave interaction theory will be, as compared 
with the correct results. Lu et al.[5] investigated the 
dependence of the wave forces of multiple bodies in 
close proximity on the incident wave frequency, the 
gap width, the body draft, the body breadth and the 
body number, based on both the viscous fluid and the 
potential flow models. Their numerical models were 
validated by the available experimental data of the 
fluid oscillation in narrow gaps. Inoue and 
Kamruzzaman[6] calculated the hydrodynamic radia- 
tion and the diffraction forces using a three-dimen- 
sional sink-source technique. In their study, the non- 
linear connecting and mooring forces were included. 
Chitrapu et al.[7] evaluated the sea keeping and maneu- 
vering performances of proximate vessels that are 
advancing forward using an efficient time domain 
method. In their study, the non-linear effects of moo- 
ring lines and fenders and the effects of viscous roll 
damping were included. Lee et al.[8] calculated the 
motions of two floating bodies in shallow water. They 
suggested that the shallow water effects be included in 
the analysis of floaters in shallow water. Kristiansen 
and Faltinsen[9] studied the coupled resonant response 
of a ship and a fixed terminal. One of their conclu- 
sions is that the linear theory over-predicts the piston- 
mode amplitude near the resonance, point about three 
times as compared with the measured value. A compa- 
rison of the hydrodynamic interactions between single 
and multi-body responses was made by Yu et al.[10].
They observed significant differences between the res- 
ponses in single and multi-body cases. 
Although the multi-body system was much stu- 
died, data accumulated are still far from enough for 
safe offloading operations in the side by side configu- 
ration in real sea states. In this study, a coupled ana- 
lysis model of the multi-body problem is built, taking 
into account the hydrodynamic interactions between 
the two vessels and the mechanical couplings between 
vessels, mooring systems, hawsers and fenders. 
Numerical simulations of the multi-body system under 
a given sea state are carried out with the help of the 
state-of-the-art code SIMO, the reliability of which 
was validated by Chen et al.[11]. To clarify the effects 
of the hydrodynamic interactions, numerical simula- 
tions are conducted ignoring the hydrodynamic intera- 
ctions, for comparison. The comparison results show 
that the hydrodynamic interactions play an important 
role in the low frequency motion responses, but have 
little effect on the wave frequency motion responses. 
The hydrodynamic interactions also have a certain 
effect on the loads acting on the hawsers and fenders. 
1. Numericalmodeling 
The numerical simulation is carried out with the 
help of the state-of-the-art time domain coupled code 
SIMO. Based on the potential theory, the hydrodyna- 
mic coefficients of the vessels in the side by side con- 
figuration, such as those of the added mass, the pote- 
ntial damping, the first-order and second-order wave 
drift forces, are calculated in the frequency domain. In 
the multi-body modeling, the hydrodynamic intera- 
ctions of the two floating bodies are also taken into 
consideration, in the form of coupled added masses 
and coupled retardation functions. 
1.1 Potential flow
The potential flows can be described by the 
Laplace equation in terms of the velocity potential 
2 2 2
2 2 2
( , , , ) ( , , , ) ( , , , )+ + = 0x y z t x y z t x y z t
x y z
� � �� � �
� � �
(1)
where ( , , , )x y z t� is the velocity potential function 
of the coordinates x , y and z , and the time t .
The velocity potential can be expressed as ( , ,x y�
i, ) = Re[ ( , , ) e ]tz t x y z �� � , in which the real part 
( , , )x y z� can further be divided into the incident 
and scattering parts 
( , , ) = ( , , ) + ( , , )I Sx y z x y z x y z� � � (2)
The scattering parts ( , , )S x y z� can further be 
divided into the diffraction part ( , , )D x y z� due to the 
existence of the floating body, and the radiation part 
( , , )R x y z� due to the oscillation of the floating body. 
And thus, Eq.(2) can be expressed as 
( , , ) = ( , , ) + ( , , ) + ( , , )I D Rx y z x y z x y z x y z� � � � (3)
Each part of the potential in Eq.(3) can be solved 
under their corresponding boundary conditions[12].
The incident and diffraction potentials would be used 
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to calculate the wave forces acting on the floating 
bodies. And the radiation potential ( , , )R x y z� would 
be used to calculate hydrodynamic coefficients such 
as those of the added masses and the damping forces 
induced by the oscillation of the floating bodies, 
0
( ) = Re dnmn m
S
a s
n
�
� � �
	 
�
� �
� �
�
 �
��
 ( , = 1, 2, , 6)m n � (4)
0
( ) = Im dnmn m
S
c s
n
�
� �� �
	 
�
� �
� �
�
 �
��
 ( , = 1, 2, , 6)m n � (5)
where m� represents the induced velocity potential 
when the floating body is oscillating in direction m
with a unit speed. ( )mna � and ( )mnc � means the 
added mass and the potential damping in direction m
induced by the body oscillation in direction n . � is 
the oscillation frequency, � is the density of the 
fluid, and 0S is the wet surface of the floating body. 
1.2 Rigid body motion
The computed frequency-domain hydrodynamic 
coefficients are used in the time-domain equation 
expressed by a two-term Volterra series expression via 
a Kramers-Kronig relation[13]. The motion equa- 
tions[14,15] in the time-domain coupled analysis for the 
FLNG vessel or the LNG carrier are formulated as 
follows 
� � � � � �
� �
� �1 2[ + ( )] + + + +M a D D f K� � � �� �� � �
� �
wave current wind ext
0
[ ( )] = + + +
t
h t F F F F� ��
�
� (6)
where M is the generalized mass matrix for the ship 
hull, ( )a � is the added mass matrix at the infinite 
frequency, K is the hydrostatic restoring stiffness 
matrix, 1D and 2D are the linear and quadratic 
damping matrices, respectively. waveF , windF and 
currentF denote the wave drag force, the wind drag 
force and the current drag force, respectively. The last 
item extF represents any other forces (the specified 
forces and the forces from station-keeping and cou- 
pling elements, etc.). ( )h � refers to the retardation 
function matrix, which is related with the influence of 
the memory effect in the free-surface and can be 
obtained by the following equation 
i1( ) = [ ( ) + i ( )]e d
2
th c a �� � �
�
��
� �
�
�
 (7)
where c and a are the radiation damping matrix 
and the added mass matrix obtained from the freque- 
ncy domain analysis, respectively, and � is the fre- 
quency. It should be noted that the damping on the 
vessel from the viscous skin drag and the wave drift 
damping should also be included in the term 
0
[ (
t
h t �
�
� �
)] d� � �� in the form of critical damping, because 
these variants are related to the motion velocity of the 
vessel. The two parts in Eq.(7) must be opposite in 
sign for 0� � and identical for 0� � , namely 
0
2( ) = ( )cos( )d =h c� � �� �
�
�
�
0
2 ( )sin ( )da� � �� �
�
�
�
�
 (8)
It should be noted that the hydrodynamic intera- 
ction effects on the frequency dependent added mass 
and the damping forces are included in the coupled 
added mass and the coupled retardation functions at 
the infinite frequency. In such a case, the 6×6 matrix 
[ + ( )]M a � in Eq.(6) should be written as a 12×12 
matrix 
, ,
, ,
( + ( )) ( ( ))
( ( )) ( + ( ))
i i i j
j i j j
M a a
a M a
� �
� �
� �
� �
� �
 (9) 
where the indices i and j refer to the FLNG vessel 
and the LNG carrier, respectively. It should be noted 
that the term with the same subscript such as ,i i or 
,j j is equal to the term in the single body case, and 
the term with the different subscripts such as ,i j
and ,j i represents the effects from the other body. 
The 6×6 impulse response function matrix [ ( )]h t ��
in Eq.(6) should be written as a 12×12 matrix 
, ,
, ,
( ) ( )
( ) ( )
i i i j
j i j j
h t h t
h t h t
� �
� �
� �
� �
� �
� �
� �
 (10)
Due to the symmetric properties, we can obtain 
the following equations 
, ,[ ( )] = [ ( )]i j j ia a� � , , ,( ) = ( )i j j ih t h t� �� �
Thus, the coupled motion equation of the two 
vessels can be expressed as a set of 12 coupled equa- 
tions[13,16]
, ,
, ,
( ( )) ( ( ))
( ( )) ( ( ))
ii i i j
j i j j j
M a a
a M a
�
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1 , 1 , 2 , 2 ,
1 , 1 , 2 , 2 ,
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
ii i i j i i i j
j i j j j i j jj
D D D D
D D D D
�
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K K
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K K
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( ) ( )
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( ) ( )
t i ii i i j
j i j j jj
Fh t h t
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� �
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�
 (11)
where iF means the forces acting on the FLNG 
vessel, which is equal to the right hand side of Eq.(6), 
while jF means the forces on the LNG carrier. In the 
numerical simulation, the integrations of the motion 
equations are carried out by using a 3rd-order Runge- 
Kutta method. 
1.3 Connection system 
The two vessels are connected through 8 hawsers 
and 4 fenders. The hawsers are modeled as linear 
springs according to 
= Tl
k
) (12) 
where l) is the elongation, k is the effective axial 
stiffness, and T is the hawser tension. 
The fender is defined as a contact element, which 
is attached to the FLNG vessel. And the fender plane 
is on the LNG carrier. The fender plane is defined by 
a point and a normal vector. The fenders are modeled 
in the same way as the hawsers, but with a lateral sti- 
ffness. 
2. Description of the side by side configuration 
The conceptual FLNG system developed by 
China National Offshore Oil Corporation (CNOOC) 
and Marine Design and Research Institute of China is 
selected as a reference. TheLNG carrier has a dis- 
placement weight of 1.6×105 t in the full loaded con- 
dition. The main parameters of the two vessels are 
listed in Table 1. The FLNG system is designed to be 
located in South China Sea at water depth of 1 500 m 
and moored by 12 mooring lines attached to an exte- 
rnal turret. The details of the mooring system are illu- 
strated in Table 2. The FLNG vessel and the LNG 
carrier are connected by 8 hawsers and 4 fenders. The 
axial stiffness of each hawser is set to be 83.57 kN/m, 
and with a Safe Working Load (SWL) of 796.95 kN 
(55% of Minimum Breaking Load (MBL) of 
1 449 kN). The force-elongation relationship of each 
fender is nonlinear. The side by side configuration of 
the FLNG vessel and the LNG carrier is illustrated in 
Fig.1. The distance between the two vessels is set to 
be 4 m, which is equal to the length of the fenders. 
Table 1 Principal Scantlings of the reference FLNG vessel 
and the LNG carrier 
Designation FLNG LNG carrier 
Length over all, oaL (m) 392.00 289.00 
Length between 
perpendiculars, ppL (m) 
356.00 278.00 
Breadth, B (m) 69.00 43.20 
Depth, D (m) 35.70 26.30 
Draft, T (m) 13.85 10.05 
Displacement, ) (t) 320 804 95 951 
Centre of gravity above 
base, KG (m) 20.553 14.49 
Centre of gravity from 
AP, LCG (m) 171.020 142.04 
Radius of roll 
gyration, xxK (m) 
25.35 14.04 
Radius of pitch 
gyration, yyK (m) 
91.36 85.13 
Radius of yaw 
gyration, zzK (m) 
93.40 86.00 
Table 2 Configuration of the mooring lines in prototype 
Designation Chain Polyester Chain 
Length (m) 100 4 000 2 000 
Diameter (m) 0.127 0.233 0.127 
weigh in air (kg/m) 322.93 33.8 322.93 
Submerged
weight (kg/m) 280.95 7.9 280.95 
Axial stiffness 
(kN/m) 1 214 733 479 000 1 214 733
Minimum 
breaking load (kN) 14 971 15 696 14 971 
Fig.1 The side by side arrangement (not in a correct scale) 
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Fig.2 Comparison of the time series of the low frequency motions of the two vessels with and without considering their 
hydrodynamic interactions 
Table 3 Summary of motion statistics of the side-by-side moored FLNG vessel and LNG carrier 
FLNG LNG carrier 
Designation
Max. Min. Mean Std. Max. Min. Mean Std. 
With interaction 0.040 –6.190 –2.85 1.04 1.540 –8.690 –3.77 2.41 
Surge 
Without interaction –0.180 –6.620 –2.85 0.90 1.790 –8.920 –4.02 2.45 
With interaction 10.20 –11.08 –1.23 6.45 14.22 –10.65 0.06 6.67 
Sway 
Without interaction 10.08 –11.71 –1.53 6.54 13.91 –11.15 –0.09 6.70 
With interaction 3.640 –3.050 0.46 1.95 4.500 –3.110 0.67 2.09 
Yaw
Without interaction 3.670 –2.940 0.54 2.00 4.720 –2.980 0.88 2.26 
During the numerical simulation, a parallel sea 
environment including wave, wind and current app- 
roaches the multi-body system with the heading of 
180o. The random wave component is described by a 
three-parameter Jonswap spectrum with a significant 
wave height of 2.5 m, a spectrum peak of 10 s and a 
peak enhancement factor of 3. A steady flow of wind 
is assumed and the mean hourly wind speed at the re- 
ference height of 10 m is 13 m/s. The current velocity 
near the free surface is 0.81 m/s. 
3. Results and discussions 
Numerical simulations of the side by side moored 
FLNG vessel and LNG carrier are carried out with and 
without considering the hydrodynamic interactions 
between the two floating vessels. Through the compa- 
rison of the numerical results, the effects of the hydro- 
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dynamic interactions of multiple bodies are revealed. 
3.1 Influence on motion responses 
The hydrodynamic interactions of the multiple 
bodies have different effects in different motion 
modes. The 6 degree-of-freedom motions can be cla- 
ssified into the following two types: the low frequency 
motions such as surge, sway and yaw, and the wave 
frequency motions such as heave, roll and pitch. 
3.1.1 Low frequency motions 
Figure 2 shows the time series of the low freque- 
ncy motions for the FLNG vessel and the LNG carrier 
in the two cases. As there are few wave frequency 
components in the low frequency motions, the time 
series and the statistics of the motion responses, in- 
stead of the spectrum analysis results, are shown in 
this section. 
Fig.3 Comparison of the time series of the relative low fre- 
quency motions of the two vessels with and without con- 
sidering their hydrodynamic interactions 
As can be seen in Fig.2 (where the surge 1( )d ,
sway 3( )d and yaw 5( )d motions are for FLNG 
vessel, and the surge 2( )d , sway 4( )d and yaw 
6( )d motions are for LNG carrier), the surge motion 
response of the LNG carrier is much larger than that 
of the FLNG vessel, while the sway and yaw motions 
see little difference. An interesting phenomenon as 
can be seen from Table 3 is that the statistics of the 
low frequency motions of each floating vessel keep 
almost the same in the two simulation cases. This 
means that the hydrodynamic interactions have little 
effects on the statistics of the motions responses. 
However, it can also be seen from Fig.2 that the 
hydrodynamic interactions change the phase of the 
low frequency motion responses. The hydrodynamic 
interactions have the most significant effects on the 
sway motion responses of the two floating vessels. 
This phenomenon is consistent with the observations 
by Buncher et al.[1]. It should be noted that the change 
of the response phase would induce a change of the 
relative motion responses. The time series of the rela- 
tive low frequency motions with and without conside- 
ring the interactions are shown in Fig.3 for compari- 
son. 
Table 4 Summary of relative motion statistics of the side- 
by-side moored FLNG vessel and LNG carrier 
Designation Max. Min. Mean Std.
With 
interaction 3.51 –4.62 –0.92 2.15Relative
surge Without
interaction 3.34 –5.12 –1.18 2.27
With 
interaction 6.09 –0.66 1.29 1.01Relative
sway Without
interaction 6.05 –0.64 1.44 1.08
With 
interaction 2.56 –2.28 0.20 0.83Relative
yaw Without
interaction 2.31 –2.33 0.34 0.95
As shown in Fig.3 (where 1r , 2r and 3r indi- 
cate the relative surge, sway and yaw motions) and 
Table 4, the hydrodynamic interactions have a signifi- 
cant effect on the phase of the relative motion respo- 
nses, but little effect on the statistics of the relative 
motion responses. It should be noted that the relative 
motions are the differences between the motion respo- 
nses of the LNG carrier and those of the FLNG vessel. 
And thus, the negative values of the relative sway mo- 
tions mean that the two vessels move towards each 
other, and correspondingly, the positive values indi- 
cate that the two vessels move away from each other. 
As there are 4 identical fenders with the length of 4 m 
between the two vessels to prevent collisions, the ne- 
gative values of the relative sway motions do not ne- 
cessarily mean collisions of the two vessels. In fact, 
when the relative distance between the two vessels is 
smaller than the fender length, the value of the relative 
654
sway motion would become negative. 
Fig.4 Comparison of the power spectrum of the motion respo- 
nses with and without considering the hydrodynamic in- 
teractions 
Fig.5 Zero-order moments of the wave frequency motions 
3.1.2 Wave frequency motions 
As the heave, roll and pitch motions contain rich 
wave frequency components, the spectrum analysis is 
carried out for these motions. As a good measure of 
the response energy, the zero-order moments of the 
wave frequency motions are calculated as the integra- 
ted area under the spectral density curve. 
Fig.6 Spectra of the tensions acting on the typical hawsers 
Figure 4 shows the power spectrum of the wave 
frequency motion responses of the two vessels with 
and without considering their hydrodynamicintera- 
ctions, where 1S and 2S indicate the power spe- 
ctrum of heave motions for FLNG vessel and LNG 
carrier. 3S and 4S indicate the power spectrum of
655
Table 5 The zero-order moments and the peak values of the spectra for the typical hawsers 
Zero-order moments(kN2) Peak values(kN2s/rad)
Serial numbers 
With Without % With Without % 
Hawser #1 909.6 714.0 27.4 123.7 58.2 112.7 
Hawser #4 2 077.0 1 487.0 39.7 246.8 121.9 102.5 
Hawser #5 1 871.0 1 804.0 3.7 229.7 169.9 35.2 
Hawser #8 1 559.0 1 133.0 37.6 245.8 121.3 102.6 
pitch motions for FLNG vessel and LNG carrier. It 
should be noted that the roll motion responses of the 
FLNG vessel are so small that no valuable parameters 
can be found. And thus, the power spectrums of the 
roll motion responses are not shown in Fig.4. To 
clarify the effects of the hydrodynamic interactions on 
the response energy of the wave frequency motions, 
the zero-order moments of the heave and pitch mo- 
tions are plotted in Fig.5, where p indicates the 
zero-order moment. A comparison of the results with 
and without the interactions shows that the hydro- 
dynamic interactions indeed have effects on the wave 
frequency motions under the given sea state, but not 
so significantly as those on the low fre- quency mo- 
tions. 
3.2 Influence on connection system 
In addition to the motion responses of the two 
vessels, the hydrodynamic interactions can also affect 
the loads on the connection systems which are related 
with the relative motion responses of the two vessels. 
3.2.1 Hawsers 
The typical connection hawsers such as hawser 
#1, hawser #4, hawser #5 and hawser #8 (whose arra- 
ngements are shown in Fig.1 in Section 2) are selected 
as the representative cases. As the time series of the 
hawser forces have rich wave frequency components, 
the power density function would be a good measure 
to analyze the hawser responses. And thus, the spe- 
ctrum analyses are carried out for the time series of 
the loads acting on the representative hawsers. 
Figure 6 shows the spectra of the tensions acting 
on the typical connection hawsers with and without 
considering the hydrodynamic interactions between 
the multiple bodies. Figure 6 shows that the conne- 
ction hawsers in different locations are acted by diffe- 
rent forces. The loads acting on the hawsers located in 
the bow of the vessels are much smaller that those in 
the stern of the vessels. The hawser #4 in the middle 
section of the vessels is acted by similar loads as those 
acting on the hawser #5. It can also been from Fig.6 
that the spectrum peak frequency of hawser #1 is simi- 
lar to that of hawser #8, and that the spectrum peak 
frequency of hawser #4 is similar to that of hawser #5. 
This means that the spectrum peak frequencies of the 
hawsers have a close relation with their locations. 
As can be seen from Fig.6, the hydrodynamic in- 
teractions have a significant effect on the responses of 
the hawser forces. Although little effect can be found 
on the spectrum peak frequencies, the peak values of 
the spectra increase sharply when the hydrodynamic 
interactions are included in the simulations (See Table 
5). To further show the effects of the hydrodynamic 
interactions, the zero-order moments, as a good mea- 
sure of the response energy of the hawser forces, are 
compared (See Table 5). It is shown that the hydro- 
dynamic interactions between the multiple bodies 
would largely increase the response energy of the 
forces acting on the connection hawsers. As shown in 
Table 5, the response energy of the forces increases by 
27.4% for hawser #1, 39.7% for hawser #4, 3.7% for 
hawser #5, and 37.6% for hawser #8. 
Fig.7 Comparison of the time series of the forces acting on the 
fenders with and without considering the hydrodynamic 
interactions 
3.2.2 Fenders 
Figure 7 (where 1f and 4f indicate the force 
acting on Fender #1 and Fender #4, respectively) and 
Table 6 show the time series and the statistics of the 
656
fender forces with and without considering the hydro- 
dynamic interactions. From Fig.7, it can be seen that 
the time series of the fender forces are in the form of 
impulses, which means that there are many collisions 
between the two vessels. In addition, the fender loca- 
ted in the bow is acted by larger loads than that in the 
stern. Figure 7 shows that the hydrodynamic intera- 
ctions can significantly affect the phase of the force 
responses. Table 6 shows that the maximum and mean 
values of the fender forces are slightly affected by the 
hydrodynamic interactions, but the standard deviation 
of the Fender #1 changes from 269.3 kN to 375 kN, 
an increase of 39.4%, due to the effects of the hydro- 
dynamic interactions. 
Table 6 Statistics of the fender forces with and without con- 
sidering the hydrodynamic interactions 
Designation Max. Min. Mean Std.
With 
interaction 2 914.4 0 115.5 375.4
Fender #1 
Without
interaction 3 051.5 0 110.5 269.3
With 
interaction 1 975.3 0 114.1 295.7
Fender #4 
Without
interaction 1 953.2 0 115.1 290.0
4. Conclusion remarks 
A numerical model for the multi-body problem is 
established with consideration of the hydrodynamic 
interactions between the two floating vessels and the 
mechanical coupling effects between vessels, mooring 
systems and connection systems. Hydrodynamic inte- 
ractions between multiple bodies are investigated 
through the comparison of the numerical results with 
and without considering the hydrodynamic intera- 
ctions. The following conclusions are obtained: 
(1) The hydrodynamic interactions between mul- 
tiple bodies can significantly affect the phases of the 
low frequency motion responses, but have little effect 
on the frequency motion responses. 
(2) Due to the existence of the hydrodynamic in- 
teractions between multiple bodies, the response 
energy of the loads acting on the hawsers would in- 
crease. And the largest increase can reach 39.7% 
under the given condition in this study. 
(3) Hawsers in different locations would be acted 
by different loads. Generally, the hawsers in the stern 
of the vessels would be acted by larger loads than 
those in the bow of the vessels. 
(4) The effects of the hydrodynamic interactions 
on the fender forces are mainly shown in the standard 
deviation of the force response. The loads acting the 
fenders are in the form of impulses, instead of conti- 
nuous series. 
This paper comes to some basic understanding of 
the hydrodynamic interactions between multiple 
bodies, focusing on their effects on the hydrodyna- 
mics of floating vessels and the connection system 
between the two vessels. Further study should be 
carried out on the relationship between the distances 
and the hydrodynamic interactions of the multiple 
bodies. 
Acknowledgement
This work was supported by the Lloyds Register 
Educational Trust (LRET) to the Joint Centre 
Involving University Coollege London, Shanghai Jiao 
Tong University and Harbin Engineering University. 
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