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Mathematical Modelling of Dynamically Positioned
Marine Vessels
Professor Asgeir J. Sørensen, Department of Marine Technology,
Norwegian University of Science and Technology,Otto Nielsens Vei 10, NO-7491 Trondheim, Norway
E-mail: [email protected]
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Outline
• Kinematics• Vessel dynamics
– Nonlinear low-frequency vessel model– Linear wave-frequency model
• Environmental loads– Wind load model– Wave load model
• Mooring system• Full-scale tests
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Pipe laying vessel
Geological survey
ROV operations
Cable laying vessel
Vibration controlof marine risers
Heavy lift operations
Position mooring
Pipe and cable laying
Dynamic Positioning and Position Mooring
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Functionality: Control Modes• Station keeping models• Marine operation models• Slender structures• Multibody operations
0 1 2 3 4 5 6 7 …..
Speed [knots]Station keeping
Marked position
Low speed tracking
High speed tracking/Transit
• Manoeuvring models • Linearized about some Uo
• Sea keeping • Motion damping
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Station Keeping ModelU 0
Low-Speed Model
-3 m/s < U < 3 m/s
Maneuvering Model
U Uo
Modelling Modelling The mathematical models may be formulated in two complexity levels:
Control plant model: Simplified mathematical description containing only the main physical properties of the process. This model may constitute a part of the controller. Examples of model based output controllers are e.g. LQG, H₂/H∞, nonlinear feedback linearization controllers, back-stepping controllers, etc. The control plant model is also used in analytical stability analysis, e.q Lyapunov Stabilty.
Process plant model: Comprehensive description of the actual process. The main purpose of this model is to simulate the real plant dynamics including process disturbance, sensor outputs and control inputs. The process plant model may be used in numerical performance and robustness analysis of the control systems.
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Kinematics - Reference Frames Earth-fixed XEYEZE - frame
The hydrodynamic XhYhZh - frame is moving along the path of the vessel. The XhYh-plane is assumed fixed and parallel to the mean water surface. In sea keeping analysis the hydrodynamic frame is moving forward with constant vessel speed U. In station keeping operations about the coordinates xd, yd, and ψd the hydrodynamic frame is Earth-fixed and denoted as reference-parallel XRYRZR - frame
(
(
)
)
X
X
E
Y
Y
E
R
x , yd d
d
V
Vc
c
ww
x , y
Body-fixed XYZ - frame is fixed to vessel body with origin located at mean oscillatory position in average water plane, (xG, 0, zG). Submerged part of vessel is assumed to be symmetric about xz-plane (port/starboard)
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Kinematics Relations
Linear and angular velocity of vessel in body-fixed frame relative to earth-fixed frame for 6 DOF - surge, sway, heave, roll, pitch and yaw:
J J0
0J
2
1
2233
3321
2
1
T2T
1 ,zyx
T2T
1 rqp ,wvu
Linear and angular vessel velocity vectors in body-fixed frame are defined:
Earth-fixed position and orientation vectors are:
X
Y
Z
u
r
v q
wp( )surge
(h eav e )
(ro ll)
(p itch )
( )ya w
( )sw ay
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Kinematics Relations
J J0
0J
2
1
2233
3321
2
1
Where J 1( ) and J2( ) are Euler rotation matrices.
X
Y
Z
u
r
v q
wp( )surge
(h eav e )
(ro ll)
(p itch )
( )ya w
( )sw ay
J1ÝR2Þ=cf cS ? sf cd+ cf sSsd sf sd+ cf cdsS
sf cS cf cd+ sdsSsf ?cf sd+ sSsf cd
? sS cSsd cScd
J11 2 J1
T 2
X2 =d%0
0
+ Cx,d
0
S%
0
+ Cx,dCy,S
0
0
f%
=J2? 1ÝR2ÞR%2
J2ÝR2Þ=1 sdtS cdtS
0 cd ? sd
0 sd/cS cd/cS
J21 2 J2
T 2
cS®0
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Vessel motion
Low-frequency Motion • Wind loads
• Current loads
• Wave loads; 2. order
• Thruster action
Wave-frequency Motion • Wave loads; 1. order
WF
LF
time
Superposition may be assumed:
Wtot F
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Nonlinear Low-frequency Vessel Model
Nonlinear 6 DOF low-frequency model - surge, sway, heave, roll, pitch and yaw :
Tccr rqpwvvuu
Relative velocity vector is defined:
M CRB CA r r D r G env moor thr
Environmental loads: Wind and 2. Order wave loads
moor Generalised mooring forces
thr Generalised thruster forces
uc =Vc cosÝKc ? f Þ, vc =Vc sinÝKc ? f Þ
env wind wave2
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Nonlinear Low-frequency Vessel Model
System inertia matrix:
M =
m ? Xu% 0 ?Xw% 0 mzG?Xq% 0
0 m ? Yv% 0 ?mzG?Yp% 0 mxG?Y r%
?Zw% 0 m ? Zw% 0 ?mxG?Zq% 0
0 ?mzG?Kv% 0 Ix?Kp% 0 ? Ixz?K r%
mzG?Mu% 0 ?mxG?Zq% 0 Iy?Mq% 0
0 mxG?Nv% 0 ? Izx?Np% 0 Iz?N r%
M =MT > 0
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Nonlinear Low-frequency Vessel Model
Generalized Coriolis and centripetal forces:
CRBÝXÞ=
0 0 0 c41 ?c51 ?c61
0 0 0 ?c42 c52 ?c62
0 0 0 ?c43 ?c53 c63
?c41 c42 c43 0 ?c54 ?c64
c51 ?c52 c53 c54 0 ?c65
c61 c62 ?c63 c64 c65 0
c41 mzGr c42 mw c43 mzGp vc51 mxGq w c52 mzGr xG p c53 mzGq u c54 Ixzp Izr
c61 mv xGr c62 mu c63 mxGp c64 Iyq
c65 Ixp Ixzr.
CRB
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Nonlinear Low-frequency Vessel Model
Generalized Coriolis and centripetal forces:
CAÝXrÞ=
0 0 0 0 ?ca51 ?ca61
0 0 0 ?ca42 0 ?ca62
0 0 0 ?ca43 ?ca53 0
0 ca42 ca43 0 ?ca54 ?ca64
ca51 0 ca53 ca54 0 ?ca65
ca61 ca62 0 ca64 ca65 0
ca42 Zw w Xw ur Zqq ca43 Ypp Yvvr Yrr
ca51 Zqq Zw w Xw ur ca53 Xqq Xuur Xw w ca54 Yrvr Krp Nrr
ca61 Yvvr Ypp Yrr ca62 Xuur Xw w Xqq ca64 Xqur Zqw Mqq
ca65 Ypvr Kpp Krr
CA r r
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Nonlinear Low-frequency Vessel Model
dNL r, r 0. 5 wLpp
DCcx r|Ucr |Ucr
DCcy r|Ucr |Ucr
BCcz r|w|w
B2Cc r|p|p zpyDCcy r|Ucr |Ucr
LppBCc r|q|q zpxDCcx r|Ucr |Ucr
LppDCc r|Ucr |Ucr
DL
Xu 0 Xw 0 Xq 0
0 Yv 0 Yp 0 Yr
Zu 0 Zw 0 Zq 0
0 Kv 0 Kp 0 Kr
Mu 0 Mw 0 Mq 0
0 Nv 0 Np 0 Nr
r atan2 vr, ur
Generalized damping and current forces:
D r DL dNL r, r
where:
Ucr ur2 vr
2
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Examples of current coefficients surge, sway and yaw for supply ship:
0 20 40 60 80 100 120 140 160 180-6
-4
-2
0
2
4
6x 10
4 Current coefficient in surge [Ns2/m2]
Degrees [deg]0 20 40 60 80 100 120 140 160 180
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
5 Current coefficient in sway [Ns2/m2]
Degrees [deg]
0 20 40 60 80 100 120 140 160 180-8
-6
-4
-2
0
2
4
6x 10
6 Current coefficient in yaw [Ns2/m2]
Degrees [deg]
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Examples of current coefficients heave, roll and pitch for supply ship:
0 20 40 60 80 100 120 140 160 180-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Current coefficient in heave [Ns2/m2]
Degrees [deg]
0 20 40 60 80 100 120 140 160 180-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5x 10
5 Current coefficient in pitch [Nms2/m2]
Degrees [deg]
0 20 40 60 80 100 120 140 160 180-2.5
-2
-1.5
-1
-0.5
0x 10
6 Current coefficient in roll [Nms2/m2]
Degrees [deg]
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Damping properties
Dominating
dampingHigh sea state Low sea state
SurgeLinear wave drift.
Nonlinear turbulent
skin friction.
Nonlinear turbulent skin friction,
when |ur | > 0.
Linear laminar skin friction,
for low KC number and ur ¸ 0.
Sway Nonlinear eddy-making.
Linear wave drift.
Nonlinear turbulent skin friction,
when |ur | > 0.
Linear laminar skin friction,
for low KC number and ur ¸ 0.
Yaw Nonlinear eddy-making.
Linear wave drift.
Nonlinear turbulent skin friction,
when |ur | > 0.
Linear laminar skin friction,
for low KC number and ur ¸ 0.
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Nonlinear Low-frequency Vessel Model
Generalized restoring forces:
G =?
0 0 0 0 0 0
0 0 0 0 0 0
0 0 Zz 0 ZS 0
0 0 0 Kd 0 0
0 0 Mz 0 MS 0
0 0 0 0 0 0
Zz ª ?_ wgAWP
ZS =Mz ª _ wg XXAW P
xdA
Kd ª ?_ wg4ÝzB ? zGÞ? _ wg XXAW P
y2dA =?_ wgVGMT
MS ª ?_ wg4ÝzB ? zGÞ? _ wg XXAW P
x2dA =?_ wgVGML
G
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Wind load :
Nonlinear Low-frequency Vessel Model
wind 0. 5 a
AxCwx w|Uwr |Uwr
AyCwy w|Uwr |Uwr
0
AyLyzCwy w|Uwr |Uwr
AxLxzCwx w|Uwr |Uwr
AyLoaCw w|Uwr |Uwr
rw u uw v vw w p q rT
uw Vw cos w , vw Vw sin w
Uwr urw2 vrw
2
w atan2 vrw, urw
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Examples of wind coefficients surge, sway and yaw for supply ship:
0 50 100 150 200 250 300 350 400-400
-300
-200
-100
0
100
200
300
400Wind coefficient in surge [Ns2/m2]
Degrees [deg]0 50 100 150 200 250 300 350 400
-2000
-1500
-1000
-500
0
500
1000
1500
2000Wind coefficient in sway [Ns2/m2]
Degrees [deg]
0 50 100 150 200 250 300 350 400-2
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
4 Wind coefficient in yaw [Ns2/m2]
Degrees [deg]
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Examples of wind coefficients heave, roll and pitch for supply ship:
0 50 100 150 200 250 300 350 400-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Wind coefficient in heave [Ns2/m2]
Degrees [deg]
0 50 100 150 200 250 300 350 400-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000Wind coefficient in pitch [Nms2/m2]
Degrees [deg]
0 50 100 150 200 250 300 350 400-2
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
4 Wind coefficient in roll [Nms2/m2]
Degrees [deg]
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2. Order Wave loads :
bwave2i =b#wm
i + bwsvi , i =1..6
=2 >j=1N Aj Tjj
i Ýg j,Kwave ? f Þ1/2
cosÝg jt + PjÞ2
Nonlinear Low-frequency Vessel Model
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Mooring SystemMooring System
OverviewOverview
XE
YE
t
X
Y
( x , y )
COT
XE
YE
Seabed
Anchor lines
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3 types of excitation:• Large amplitude LF motions• Medium amplitude WF motions• Very high frequency vortex-induced vibrations
T T T
H H H
~
~
H
T
TP
Single Line ModellingSingle Line Modelling
Mooring SystemMooring System
Xh
Seabed
Z
X
D
Ls
H
xh
Ltot
TP
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T f X
H f XT h
H h
( )
( )
Mooring SystemMooring System
Line CharacteristicsLine Characteristics
Xh
Seabed
Z
X
D
Ls
H
xh
Ltot
TP
1370 1380 1390 1400 1410 1420 14300
1000
3000
5000
7000
Horizontal distance to anchor [m]
T
H
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n
1iiiiiii
ii
ii
mo
cosyHsinxH
sinH
cosH
g
x x x y
y y x y
iTPE
t iTP
t iTP
iTPE
t iTP
t iTP
cos( ) sin( )
sin( ) cos( )
0 0
0 0
h x yi ih
ih ( ) ( )2 2
x x x
y y y
ih
iA
iTPE
ih
iA
iTPE
i ih
ihy xatan2( , )
x x x
y y y
i iTPE
i iTPE
• Additional damping term • Restoring term
mod
mog
Forces and moment on moored structureForces and moment on moored structure
Mooring SystemMooring System
XE
YE
i
xiTPE, yi
TPE( )
xih
yih
Turret
TP
Anchor
hi
xiA( ),yi
A
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Quasi-static mooring model:- use the line characteristics for each line i in
H f hi Hi i ( )
Mooring SystemMooring System
Generalized mooring forces in LF modelGeneralized mooring forces in LF model
(.)g)(J(.)d moT
momo
n
1iiiiiii
ii
ii
mo
cosyHsinxH
sinH
cosH
g
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Mooring SystemMooring System
Linearized Mooring ModelLinearized Mooring Model
H H c hi i i i 0 cdf
dhh hi
Hi
ii i ( )0
)(G)(g)(|g
)(gg 00mo0mo
0momo 0
Mmo Dd
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Linear Wave-frequency Vessel ModelPotential theory is assumed, neglecting viscous effects.
Two sub-problems:• Wave Reaction: Forces and moments on the vessel when the vessel
is forced to oscillate with the wave excitation frequency. The hydrodynamic loads are identified as added mass and wave radiation damping terms.
• Wave Excitation: Forces and moments on the vessel when the vessel is restrained from oscillating and there are incident waves. This gives the wave excitation loads which are composed of so-called Froude-Kriloff (forces and moments due to the undisturbed pressure field as if the vessel was not present) and diffraction forces and moments (forces and moments because the presence of the vessel changes the pressure field).
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Linear Wave-frequency Vessel Model
Linear 6 DOF Wave-frequency model - surge, sway, heave, roll, pitch and yaw :
Earth-fixed motion vector:
1. Order wave loads
M Rw Dp Rw G Rw wave1
w J 2Rw,
Rw R6
w R6
Motion vector in hydrodynamic frame:
2 0 0 d T
wave1 R6
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Verification tests on Varg FPSOVerification tests on Varg FPSO
Seabed
Anchor lines
Turret
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0 100 200 300 400 500 600700
750
800
850
900
950
1000
1050
1100
1150
Measured tension, line 1-5 [kN]
0 100 200 300 400 500 600650
700
750
800
850
900
950
1000
1050
1100
1150Measured tension, line 6-10 [kN]
time [sec]
0 100 200 300 400 500 600
-50
-45
-40
-35
-30
-25
-20
Measured wind direction [deg]
0 100 200 300 400 500 60018
19
20
21
22
23
24
25
26
27
Measured wind velocity [m/s]
time [sec]
Full-scale results, Varg FPSOFull-scale results, Varg FPSO
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Full-scale results, Varg FPSOFull-scale results, Varg FPSO
0 100 200 300 400 500 600-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Estimated (blue) and measured (red) North position [m]
0 100 200 300 400 500 600-2
-1
0
1
2
3
4
Estimated (blue) and measured (red) East position [m]
0 100 200 300 400 500 60053.5
54
54.5
55
55.5
56
56.5
57Estimated (blue) and measured (red) heading [deg]
time [sec]
0 100 200 300 400 500 600
-600
-400
-200
0
200
400
600
800
1000
Measured (red) RPM, thruster 1
0 100 200 300 400 500 600
-600
-400
-200
0
200
400
600
800
1000Measured (red) RPM, thruster 2
0 100 200 300 400 500 600
-600
-400
-200
0
200
400
600
800
1000Measured (red) RPM, thruster 3
time [sec]
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