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1 INTRODUCTION
Manoeuvring simulations are a crucial tool for
evaluating the safety and viability of operations
carried at the sea without exposing personnel or
assets to risk. On a manoeuvring simulator, the
simulated operations can vary widely from navigation
in access channels to underway ship-to-ship
offloading. In this context, a maritime simulator must
incorporate elements usually found in those scenarios
and simulate the operation and behaviour of the
vessels with sufficient realism.
To achieve the required level of realism, a
numerical simulator integrates, in real-time, a
complex mathematical model. The numerical
simulator computes the floating body dynamics
considering hydrodynamic effects and interactions
between the vessels and simulated elements such as
mooring lines and environmental forces.
One of such interaction is the contact between
bodies, for example, between a vessel and the marine
fenders in a berth or when a tugboat pushes a ship to
bring it closer to the final position. This physical
interaction is complex and depends on the mutual
deformation of the bodies and surface interactions
(friction). To incorporate such phenomenon in the
simulation, a simplified model of the contact is
needed.
The modelling of marine fenders and its
interactions with vessels and other structures is a vast
field, encompassing materials science, analysis made
by numerical and empirical models, and optimization
of fender properties.
Some research sources provide simplified
analytical or empirical approaches for calculating
fender forces and other useful parameters [1-2]. Those
approaches are usually focused in providing tools for
dimensioning piers, berths and the fenders
themselves.
Other works focus on the calculus and
optimization of fender properties, such as energy
absorption, restitution coefficients, etc. by the use of
finite element analysis and other numerical models
Vessel-Fender Contact Force Modelling for a Real-time
Ship Manoeuvring S
imulator
F.
M. Moreno, H.A.U. Sasaki, H.S. Makiyama & E.A. Tannuri
University of São Paulo, São Paulo, Brasil
ABSTRACT: This paper presents the development of a method for calculating the horizontal contact forces
between two bodies in a real-time ship manoeuvring simulator. The method was implemented in the simulator
of the University of São Paulo, whose computing core is named “Dyna”. The model proposed calculates
restoration and friction forces between bodies and has a Momentum-Imp
ulse based criterion to reduce
numerical issues when the simulation numerical integration has large time-steps. The model was empirically
evaluated at the simulator by deck officers, in real-time simulations with pilots and tugmasters. We also ran
simulations of that model to compare its performance under different integration time-steps lengths.
http://www.transnav.eu
the
International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 17
Number 4
December 2023
DOI: 10.12716/1001.17.04.
07
814
for solving the deformable body equations [3-5].
Despite being valuable tools, this approach is
currently unpractical for Real-Time applications due
to the computational costs required.
This paper presents a contact force model
developed and implemented in a Real-Time Ship
Manoeuvring Simulator. The model separates the
contact forces into restoration and friction forces. It
adopts a Momentum-Impulse balance criterion to
allow its use in a numerical simulator with large time-
step integration without numerical issues.
The next section provides an overview of the
proposed Contact Engine, section 3 explains the
model implemented, section 4 compares the model
behaviour for different time-step integrations, and
section 5 presents the conclusion.
2 OVERVIEW
The proposed model was developed for the TPN Ship
Manoeuvring Simulation Center. That simulator
adopts a 6-DoF mathematical model (TPN-MM) based
on a long-term experience in ship hydrodynamics.
Besides, the TPN-MM follows ITTC procedures to
calibrate and validate manoeuvring models. The TPN-
MM is based on a quasi-explicit heuristic model for
estimating manoeuvring forces on ships. It also
comprises models for environmental forces (wind,
current, waves), restricted water effects (bottom and
bank interaction) and external forces (cables, vector
tugboats). The numerical code has been described in
detail in the references [6]. It can also incorporate
forces provided by external programs, such as the
Contact Engine presented in this paper.
Despite the Manoeuvring Simulator having 6 DoF,
the proposed Contact Engine considers only forces
and geometries of contacting bodies in the 3 DoF
horizontal plane. This approach was adopted to
simplify the model and reduce computational cost
without compromising the realism of the simulation,
since vertical contacting forces are negligible during
regular operations.
That 2D Contact Engine should provide as output
the contact forces that will be integrated in the
dynamical simulation with a time-step of Δt=0.1s
second by forward Euler. The Contact Engine receives
as inputs the 2D geometric shape of the objects in the
simulation scenario, their current positions and
velocities, and the physical contact parameters
defined by the user. An overview of the process is
shown in the flowchart in Figure 1.
3 METHODOLOGY
This section explains the simplified method adopted
for calculating the interaction force between two
objects. The next subsections, from 3.1 through 3.3,
explain how the forces are calculated. Section 3.4
describes how an impulse-momentum heuristic is
applied to limit the calculated forces in order to avoid
numerical issues.
Figure 1. Flowchart of the process for computing contact
forces in the Contact Engine and how it interacts with the
numerical simulator
3.1 Definitions
The model considers the contacting objects as being
2D convex polygons in the horizontal plane
OXY
(Figure 2) which have positions and velocities defined
externally by the manoeuvring simulator. Each object
represents a vessel, harbour fender or another port
element. The model is defined in discretized time, and
it runs with the same time-step of the manoeuvring
simulator (Δt=0.1s). From here on, the superscript n
will be used to denote the time-step of the variables
and the subscript pair i,j will denote that the variable
is respective to object j and the direction i.
Figure 2. Definition of components obtained from the
contact between two objects. It is also worth noticing that
under normal circumstances, the intersection area A is
much smaller than the one depicted in the Figures in this
work.
815
The contact happens when there is an intersection
between two object polygons. The contact polygon S
is defined as the intersection between the projected
area of the colliding objects in the plane OXY and its
centroid being defined as C (Figure 2). It is also
defined the vector
j
r
as the distance of the C point
from the barycentre of each object B
j.
The intersection of the two polygon edges also
produces two intersection points (P
1 and P2) that are
used to define a “contact line” between the bodies
(Figure 1b). The contact line is used to define a local
base
oxy
, with origin in C, with axis
⊥
12
x PP
and
12
y PP
, and unitary vectors
ˆ
i
and
ˆ
j
respectively.
Figure 3. Intersection points and “contact line” (red dotted
line) and contact basis ¯oxy for the time-step n
These definitions are kept as the system evolves in
time. The contact forces are divided into two
components, a normal one N
n
and a tangential one T
n
and both have their direction set as
⋅=
ˆ
0
n
n
Nj
and
⋅=
ˆ
0
n
n
Ti
respectively, as shown in Figure 3.
As these forces are equally applied in the two
bodies by the third Newton’s Law (Figure 4) [1], the
following relation (Equation (1)) can be established:
{ }
=
= −
=
,
,
, 1,2
nn
ij
F TN
F F with
ij
(1)
Figure 4. Forces applied on both objects in time-step n. Force
is applied at the point C
n
.
We assume that forces in the
i
direction are
provoked by elastic deformation of the bodies, and
forces in the
j
direction are due to friction between
the bodies. We will consider that the contact forces are
solely applied in the point C on both objects.
Each object is considered a rigid body and has an
associated mass m
j and rotational inertia Ij around the
body barycentre B
j. The mass mj is the sum of both
inertial and added mass in the
OXY
directions.
Since the value of added mass changes with the
direction of acceleration, only the smallest value is
considered in m
j. In cases of harbour fenders or other
fixed objects, we consider that
→∞
j
m
and
→∞
j
I
.
3.2 Normal Contact Model
The model’s normal forces are obtained from a spring-
dampener model defined in equation (2), where k is
an area stiffness and c is a damping coefficient. The
model is discretized in time by a first-order, backward
finite difference, thus obtaining the equation (3). The
dot product is defined by the operator (·) and
( )
sgn
is the sign function.
( ) ( )
( )
( ) ( )
( )
( )
=− ⋅ −⋅ ⋅ ⋅
ˆˆ
sgn
jj
dS t
N t kSt c r t it it
dt
(2)
−
−
=− ⋅ −⋅ ⋅ ⋅
∆
1
ˆˆ
sgn
nn
n
n nn
jj
SS
N kS c r i i
t
(3)
3.3 Tangential Contact Model
Regarding the calculation of the tangential force
n
T
,
the collision polygon was considered a soft fender, as
exemplified in Figure 5a. This soft fender is held in
the second body by friction forces in a reference point
P
F, and it can be sheared if there is relative movement
between the vessels in the
ˆ
n
j
direction, as shown in
Figure 5b. The amount of Shearing, or lateral
displacement, is defined as Δs
n
.
Figure 5. Fender model representation. The fender can be
sheared if there is relative motion between contacting
objects.
That shearing produces a restoration force that
tends to force both objects to return to their initial
relative positions in the
ˆ
n
j
direction. The magnitude
of those restoration forces is proportional to both the
shearing of the fender in the
ˆ
n
j
direction (Δs
n
), and a
shearing stiffness k
s, as shown in equation (4).
=−∆
1
ˆ
n
nn
s
T k sj
(4)
Equation (4) is valid if there is no sliding between
the fender and the contacting object in the time-step n.
We can consider that there is no sliding if the static
friction force is enough to counteract
n
i
T
. This
condition is satisfied when the inequality in equation
816
(5) is satisfied, where μ
s is the coefficient of static
friction between the object and the fender.
µ
∆≤
n
n
j
ss
ks N
(5)
If the restoration forces are larger than the static
friction force, then the fender slides in the direction of
Δs
n
. If there is sliding, the restoration force is limited
by the dynamic friction force defined in equation (6),
where μ
k is the coefficient of dynamic friction between
the object and fender.
(
)
µ
=−∆
1
sgn
nn
n
j
k
TN s
(6)
We consider that when the fender slides in one
time-step it will slide enough to achieve equilibrium
in the next time-step (Figure 6). Thus, in the next time-
step, we update the contact point considering a
residual fender shearing that is enough to equilibrate
the dynamic friction, as defined in equation (7).
( )
( )
µ
µ
+
∆ = ∆⇒ ∆
1
sgn sgn
n
j
k
n
nn n
j
sk
s
N
ks N s s
k
(7)
Figure 6: Fender sliding along the contact line. The sliding
happens in one time-step and the fender is still sheared after
the sliding stops.
We can summarize the fender behaviour in
equation (8), where the fender has two states
depending on the amount of shearing force: for small
enough displacements, the fender point P
F is kept
fixed in the object by static friction, while for larger
displacements, it slides along the contact line.
( )
( )
µ
µ
µ
µ
+
+
=−∆ ∆≤
=− ∆ ∆>
∆= ∆
1
1
1
1
ˆ
()
sgn ( )
sgn
n
j
s
n
nn t
s
s
n
j
s
nn
nt
j
k
s
n
nn
k
j
s
N
T k s j if s NoSliding
k
N
T N s if s Sliding
k
s sN
k
(8)
To apply that fender model to our collision model,
we need a heuristic to calculate the lateral
displacement Δs
n
. We achieve this by following the
steps below:
1. We define the lateral displacement at the start of
the contact as zero (Δs
0
=0).
2. While there is contact between the bodies:
− Consider two points coincident to C
n
that are
individually fixed in objects 1 and 2 at the time-
step n=T≥0, here denominated
12
,
TT
CC
.
− At the time-step n=T+1 we compare how much
1
T
C
moved in relation to
2
T
C
(Figure 7) along
the direction of
+1
ˆ
n
j
and add the value to Δs
T
,
following the formula
( )
++
∆ =∆+ − ⋅
11
21
T T T TT
s s CCy
.
− If there was sliding in the previous time-step,
n=T, define Δs
T+1
accordingly to equation (8).
Figure 7. Depiction of how the amount of fender shear is
computed in our model, the point C
T
is fixed in both objects
at one time-step, then the amount of displacement between
those points in the next time-step is used as basis for the
computation fender shear.
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3.4 Impulse-Momentum Approach for Limiting Contact
Forces
In situations where the contact between objects has
high stiffness (large k), or the relative velocity
between objects is large, there is the potential to
emerge unrealistic large contact forces that will
produce numerical issues unless a very small
integration time-step is used. In order to avoid such
issues in the simulator, we use an Impulse-
Momentum heuristic to limit those contact forces.
We consider that the contact is a partially elastic
collision [1]. Therefore, after the contact, the objects
will have a relative velocity that is at most a fraction
of the relative velocity at the start of the contact, as
defined by (9). In this equation,
0
x
VC
is the initial
relative velocity in the
0
ˆ
i
direction,
0
y
VC
is the
initial relative velocity in the
0
ˆ
j
direction, γ is the
restitution coefficient of the contact, and
final
T
x
VC
and
final
T
y
VC
are the relative velocities at the end of the
contact between objects.
γ
γ
≤−
≤−
0
0
final
final
T
xx
T
yy
VC VC
VC VC
(9)
The relative velocities
n
x
VC
and
n
y
VC
are
defined as the difference between the instantaneous
velocities of points
2
n
C
and
1
n
C
, as shown in
equation (10), where,
1
n
C
V
is the absolute velocity of
the point
1
n
C
, and
2
n
C
V
is the absolute velocity of
the point
2
n
C
( )
( )
=−⋅
=−⋅
12
12
ˆ
ˆ
n n nn
x CC
n n nn
y CC
VC V V i
VC V V j
(10)
One way to guarantee that the inequalities in
equation (9) are satisfied is to limit the change of
velocity in one time-step to a value that will not make
the velocity in the next time-step violate those
inequalities. From now on we will adopt that
criterion, described in equation (11), where the delta
operator (Δ) denotes the variation in the quantity
from the current time-step to the previous.
γ
γ
+
+
∆=−
∆=−
10
10
nn
x xx
nn
y yy
VC VC V
VC VC V
(11)
And by applying the delta operator to equation
(10) we can relate the change in relative velocity
+
∆
1n
i
VC
to the change of velocity of points
1
n
C
V
and
2
n
C
V
as shown in equation (12).
( )
( )
+
+
∆ = ∆ −∆ ⋅
∆ = ∆ −∆ ⋅
1
12
1
12
ˆ
ˆ
n n nn
x CC
n n nn
y CC
VC V V i
VC V V j
(12)
The relation between the change in velocity and
the applied force is defined by the Impulse-
Momentum balance equations (13) [1], where the first
equation links the linear velocity variation of the body
barycentre
(
)
∆
J
B
v
with the sum of applied forces in
the body
(
)
∑
J
F
, and the second one links the
angular velocity
(
)
ω
∆
J
with the sum of applied
torques
( )
τ
∑
J
. m is the mass and I is the moment of
inertia of the body. Δt is the duration of the
application of force or torque. The applied torque and
moment of inertia are referent to the body barycentre.
ωτ
∆= ∆
∆= ∆
∑
∑
J
jB J
jJ J
mv Ft
It
(13)
To use equation (13), we must link the relative
velocities between objects and the velocities of the
objects barycentre. This can be achieved by
considering that both objects are rigid bodies, so we
can obtain the
J
n
C
V
velocities from the velocities of
the objects at their barycentre (
J
n
B
V
), plus their
angular velocities,
ω
n
J
, and the arm vector that
connects the
n
C
to the object respective barycentre
t
J
r
(as in Figure 2), by using the rigid body formula
(14) [2].
ω
= +×
JJ
n n nn
C B JJ
VV r
(14)
We can define that the variation of
J
n
C
V
and
J
n
C
V
from one time-step to the next depends on the
changes of barycentre (
J
n
B
v
) and angular velocity
(
ω
n
J
) as shown in equation (15). For simplicity, we
assume that the vectors
n
J
r
are kept almost constant
from one time-step to the next.
ω
++ + +
∆ = − =∆ +∆ ×
11 1 1
J JJ J
n n n n nn
C C C B JJ
V VV V r
(15)
Considering the Impulse-Momentum balance
equation (13), we can rewrite the rigid body equation
(14) and obtain the equation (16), considering the
normal and parallel contact forces actuating on the
object.
(
)
τ
ω
+
∆ =∆ +∆ × = ∆ + ∆ ×
∑
JJ
nn
n
JJ
J
n n nn n
C B JJ J
jj
TN
V V r t tr
mm
(16)
The sum of contact torques applied on object 1
(
τ
∑
1
can be obtained from the sum of contact forces
and the arm between C
A and the object barycentre, as
shown in equation (17):
( )
τ
=×+
∑
nn n n
JJ J J
r TN
(17)
And by substituting equation (17) on (16) we
obtain equation (18):
( )
( )
+
×+ ×
+
∆ = ∆+ ∆
1
J
nnn n
nn
JJ J J
JJ
n
C
jj
r TN r
TN
Vt t
mI
(18)
By applying the vector triple product, it is possible
to decompose the cross product in equation (18) and
obtain equation (19):
818
( )
( ) ( )
+
+
∆
∆ = ∆+ + − ⋅ +
2
1
J
nn
JJ
n n nnnnnn
C JJJJJJJ
jj
TN
t
V t r TN rTNr
mI
(19)
And by decomposing the change in velocity
+
∆
1
J
n
C
V
in equation (19) into
ˆ
n
i
and
ˆ
n
j
components
we obtain equation (20):
( )
( ) ( ) ( )
( )
( ) ( ) ( )
+
+
∆
∆ = ∆+ − ⋅ + ⋅
∆
∆ = ∆+ − ⋅ + ⋅
,
,
2
1
2
1
ˆˆ
ˆˆ
Jx
Jy
n
J
t n n n n n tnn
C J J JJ J J
jj
n
J
t n n n n n tnn
C J J JJ J J
jj
N
t
V t r N r T N ri i
mI
T
t
V t r T r T N rj j
mI
(20)
The equation (20) associates the change in velocity
with the collision forces, but the presence of both
n
j
T
and
n
j
N
in the same equation poses a problem, since
no explicit formula can be obtained for all cases. The
solution adopted in this work considers that the forces
normal to the contact line are higher than the parallel
forces
( )
nn
jj
NT
, so the equation for the
+
∆
,
1
Jx
n
C
V
component can be simplified by discarding the
n
j
T
force, as shown in equation (21):
( )
( )
( )
( )
++
∆
⋅= ⋅
∆
∆ = + −⋅ ⋅ ∆ =
⇒
∆∆
= + −⋅ =∆ +
,
2
11
2
2
ˆ
ˆˆ
1
ˆ
J Jx
n
n n nnn
j
n nnnnnnn t
J j jJ
C J jJ jJ C
jj
nn
jj
n tn n
JJ j
jj j
Nt
r N Nri
t
V rNrNrii v
mI
r
NtNt
r ri N t
mI m
( )
−⋅
2
2
ˆ
n tn
JJ
j
ri
I
(21)
A similar approach is used for the
y
component
+
∆
,
1
Jy
t
C
V
, where by applying the distributive property
to equation (19) it is possible to rearrange it and
obtain an equation similar to (21), as shown below in
equation (22). For the computation of change in
velocity in
ˆ
n
j
it is considered the equation with both
force components
( )
+
nn
jj
TN
, and thus it is obtained
an additional term that depends on the
n
j
N
force.
( )
( )
( ) ( ) ( )( )
+
∆
∆ = ∆+ − ⋅ + ⋅ ⋅ =
∆
=∆+ −⋅ −⋅ ⋅ =
=
,
2
1
2
2
ˆ ˆ ˆˆ
ˆ ˆˆˆ
Jy
n
j
t n n nn n tn n nnn
C J j J jJ jJ
jj
n
j
n n nn n nn nn nn
J j J jJ J j
jj
n
j
T
t
V t r T rjT rjN rjj
mI
T
t
t r T rj T ri rj Tj
mI
T
( ) ( )( )
∆
∆ + −⋅ − ⋅ ⋅
2
2
11
ˆ ˆˆˆ
n nn nn nn nn
JJ J J j
jj j
t
t r rj ri rj Tj
mI I
(22)
We also further define
n
xj
E
and
n
yj
E
as the
composition between mass, rotational inertia and
torque arm of the bodies, as shown by the equation
(23). This component can be seen as a value of
“effective” mass, and it dictates the resistance to
velocity change in the point C
n
of the object when a
force is applied at that point.
( )
( )
−
−
−⋅
= +
−⋅
= +
1
22
1
22
ˆ
1
ˆ
1
n nn
JJ
n
xj
jj
n nn
JJ
n
yj
jj
r ri
E
mI
r rj
E
mI
(23)
By substituting equation (22) for both objects 1 and
2 in equation (12), we obtain equation (25) that
associates the contact force with the change in relative
velocity in the
ˆ
n
i
direction.
( )
( )
12
11
12
12
1
12
12
nn
nn
nn
xx
n nn
xx
nn
nn
xx
xx
EE
NtNt
NN
VC VC tN
EE
EE
++
+
∆∆
= −
∆=− ∆=∆
⇒
(24)
By applying the same approach for the
ˆ
n
j
direction, the equation (25) is obtained, where for
brevity, we define L
n
as being the second term of the
equation, as shown in equation (25):
( )( )
( )( )
( )
( )
+
+
∆∆
∆∆
∆= −⋅⋅ − −⋅⋅
= −
∆
⇒
+
=∆ +∆ ⋅
1
12
11 1 22 2
12
12
1
12
12
1 12
2
12
ˆˆˆ ˆˆˆ
1
ˆ
y
y
nn
n nn nn nn nn nn nn
C
nn
yy
nn
n
C
nn
yy
n n nn
nn
yy
Tt Tt
tt
V ri rj Tj ri rj Tj
II
EE
TT
V
EE
tT t N r i
I
EE
( )
( )( )
( )
⋅− ⋅ ⋅
+
=∆+
2 11
1
12
1
12
1
ˆ ˆ ˆˆ
nn nn nn n
nn
yy
nn
nn
yy
rj ri rj j
I
EE
tT L
EE
(25)
And by substituting equations (24) and (25) in the
criterion equation (11) we finally obtain equation (26),
where the maximum contact force for both directions
is obtained. The normal and parallel forces calculated
by equations (3) and (8) are limited to the maximum
value obtained from equation (26). The forces for
object 2 can be obtained by considering that they are
the reaction forces of
1
n
T
and
1
n
N
.
( )
(
)
( )
( )
( )
(
)
( )
( )
1 2 12
00
11
12
12
1 2 12
00
11
12
12
1
1
n n nn
x x xx
n n nn
xx xx
nn
nn
xx
xx
n n nn
y y yy
n tn n n n
xx xx
nn
nn
yy
yy
E E EE
tN VC VC N VC VC
t
EE
EE
E E EE
tT J VC VC T VC VC L
t
EE
EE
γγ
γγ
+
∆ ≤ − ⇒≤ −
∆
+
+
∆ +≤ − ⇒ ≤ − −
∆
+
(26)
4 VALIDATION
We have evaluated the performance of the Contact
Engine in two experiments, in the first one
simulations run with integration time-steps varying
from 0.1s to 2.0s for both the Contact Engine and the
numerical engine are evaluated. In the second
experiment the relative velocity of the bodies at the
start of the simulation is varied from 0.5 knot to 4.0
knots.
For those evaluations it was considered a scenario
with two vessels as objects as depicted in Figure 5.
The object 1 is a container vessel 298m long with 46m
beam, 9m draft and displacement 87.800 tons. The
object 2 is a tugboat 32m long with 13m beam and
displacement 930 tons. The object 2 starts the
simulation with zero velocity and object 1 starts with
a velocity of 4 knots ahead. The objects are 4m apart at
the start of simulation
819
Figure 8. Initial condition for the experimental test. Object 1
and 2 are separated by 1.7m.
The simulations are run in a steady sea, without
environmental conditions (wind, waves and currents),
and relative velocities between objects and the normal
contact forces are written to a text report. As an
example we have used the parameters with values
defined on Table 1. But it is worthwhile to point that
those parameters can be configured to represent the
properties of specific fenders if needed.
Table 1. Parameters used for the contact between objects in
the experimental setup.
________________________________________________
Variable Value
________________________________________________
K 500 kN/m²
C 0 kN.s/m
K
s 3000 kN/m
μ
s 0.5
μ
k 0.5
γ
0.1
∆t (1st experiment) [0.1,0.5,1.0,2.0] s
∆t (2nd experiment) 0.1 s
________________________________________________
The results for contact normal forces are shown in
Figure 9, where it is possible to notice that smaller
time-steps produce higher forces with shorter
durations, while larger time-steps produce a smaller
force distributed over a longer period. This behaviour
is expected since the Impulse-Momentum criterion
enforces that the integral of force and time is kept the
same independent of the time-step length used in the
numerical engine.
Figure 9. Normal contact forces produced in the
experimental setup for varying integration time-steps.
Also is possible to notice in Figure 9 that as the
time-step decreases the force peaks tend to converge
to a value (eg. the difference between peaks for 0.5s
and 0.1s ∆t is smaller than the difference between
peaks for 0.5s and 1.0s ∆t) since the discretized model
tends to approach the true solution as the time-step
length is shortened.
The relative velocity between objects is shown in
Figure 10. It is possible to notice that for all time-steps,
the Contact Engine produced similar results, with
objects ending the contact with a residual velocity that
is similar to the expected theoretical value. The only
notable difference is that for larger time-steps the
contact takes longer and the relative velocity
converges slower to the expected value.
Figure 10. Relative velocities between objects during the
contact. The dotted line is the final velocity expected by the
equation (10).
To evaluate the effect of initial relative velocity, the
same scenario was used for the second experiment,
but the initial velocity of object 2 was modified
between 0.5 knot and 4 knots and the distance
between vessels has been reduced to 2 meters, and
thus the results in Figure 11 were obtained, where it is
possible to notice that the same behaviour is obtained
for different contact speeds.
Figure 11. Relative velocities between objects during the
contact for different initial velocities for the body 2.
The forces during the contact also behave as
expected, with smaller peak forces for collision with
820
lower velocity between the vessels. The duration of
the first contact forces peak also slightly increases for
contacts in lower velocities, having a duration
between 1 and 2 seconds. It is also relevant to notice
that the intersection between vessel polygons is kept
small for all the contact velocities, as shown in table 2,
having a maximum penetration between geometries
of 0.97m for the 4 knots scenario.
Figure 12. Contact forces for scenarios with different relative
velocities between objects.
Table 2. Maximum penetration between vessels’ polygons.
________________________________________________
Initial velocity Maximum depth
________________________________________________
4.0 knots 0.96 m
2.0 knots 0.53 m
1.0 knot 0.29 m
0.5 knot 0.14 m
________________________________________________
5 CONCLUSIONS
In this paper, a 2D time-discretized model for contact
forces between bodies was presented. Its performance
was evaluated by comparing its behaviour for varying
integration time-steps duration in a numerical
manoeuvring simulator. The contact model performs
close to the expected, considering the hypothesis
adopted in its formulation, even for large integration
time-steps.
The model considers both restoration forces and
friction forces between bodies. The model is
simplified, so it can be run in Real-Time simulations,
and can be satisfactorily used for cases where the
dynamical behaviour of the bodies in the simulation is
more important than the accurate computation of
peak values or contact forces.
As further steps for improving the model, one
possibility is to run high-resolution Finite Element
Analysis of the contact between typical marine
fenders and ship hulls, and use it as a gold standard
to compare to the model proposed. A direct
comparison between the forces calculated by both
models will provide a way to objectively measure the
performance of the proposed model, and it might give
insight into where the model could be improved.
ACKNOWLEDGMENTS
The authors acknowledge ANP/Petrobras for sustaining
long-term support in the development of the Ship
Maneuvering Simulator Center (SIGITEC 2018/00402-5).
The first and last authors also thank the Center for Artificial
Intelligence (C4AI-USP), supported by the São Paulo
Research Foundation and by the IBM Corporation (FAPESP
grants 2019/07665–4 and 2020/16746-5).
The last author acknowledges the CNPq – Brazilian
National Council for Scientific and Technological
Development for the research grants (process 310127/2020–
3).
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