1 Mathematical Modelling of Dynamically Positioned Marine Vessels Professor Asgeir J. Sørensen,...

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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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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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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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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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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.


when |ur | > 0.



Yaw Nonlinear eddy-making.

Linear wave drift.


when |ur | > 0.



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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 :


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


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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


Xh

Seabed

Z

X

D

Ls

H

xh

Ltot

TP

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T f X

H f XT h

H h

( )

( )


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


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 ( )


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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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

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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]

1 Mathematical Modelling of Dynamically Positioned Marine Vessels Professor Asgeir J. Sørensen,...

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