1. 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

2. Outline Kinematics Vessel dynamics Environmental loads Mooring system
Nonlinear low-frequency vessel model
Linear wave-frequency model
Environmental loads
Wind load model
Wave load model
Mooring system
Full-scale tests

3. Dynamic Positioning and Position Mooring
Pipe and cable laying
Vibration control
of marine risers
Position mooring
ROV operations
Heavy lift
operations
Geological survey
Cable laying vessel
Pipe laying vessel

4. Functionality: Control Modes
Station keeping models
Marine operation models
Slender structures
Multibody operations
Manoeuvring models
Linearized about some Uo
Sea keeping
Motion damping
High speed tracking/Transit
Low speed tracking
Marked position
Station keeping
Speed [knots] 1 2 3 4 5 6
7 …..

5. 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.
Station Keeping Model
U  0
Low-Speed Model
-3 m/s < U < 3 m/s
Maneuvering Model
U  U o

6. 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
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)

7. 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:
Earth-fixed position and orientation vectors are:
Linear and angular vessel velocity vectors in body-fixed frame are defined:

8. Kinematics Relations
Where J 1( ) and J2( ) are Euler rotation matrices. J ? 1 Ý R Þ = J T Ý R Þ J ? 1 Ý R Þ J T Ý R Þ 1 2 1 2 2 2 2 2

9. Vessel motion Low-frequency Motion Wave-frequency Motion Wind loads
Current loads
Wave loads; 2. order
Thruster action
Wave-frequency Motion
Wave loads; 1. order
Superposition may be assumed:

10. Nonlinear Low-frequency Vessel Model
Nonlinear 6 DOF low-frequency model - surge, sway, heave, roll, pitch and yaw : M X % + C Ý X Þ X + C Ý X Þ X + D Ý X Þ + G Ý R Þ = b + b + b RB A r r r
env
moor
thr
Relative velocity vector is defined: b = b + b
Environmental loads: Wind and 2. Order wave loads
env
wind w a v e 2 b
Generalised mooring forces
moor b
Generalised thruster forces
thr

11. Nonlinear Low-frequency Vessel Model
System inertia matrix:

12. Nonlinear Low-frequency Vessel Model
Generalized Coriolis and centripetal forces: C Ý X Þ X RB c = mz r c = mw c = m Ý z p ? v Þ 4 1 G 4 2 4 3 G c = m Ý x q ? w Þ c = m Ý z r + x p Þ c = m Ý z q + u Þ c = I p ? I r 5 1 G 5 2 G G 5 3 G 5 4 xz z c = m Ý v + x r Þ c = ? mu c = mx p c = I q 6 1 G 6 2 6 3 G 6 4 y c = I p + I r . 6 5 x xz

13. Nonlinear Low-frequency Vessel Model
Generalized Coriolis and centripetal forces: C Ý X Þ X A r r c = ? Z w ? X u ? Z q c r = Y p + Y v + Y r a 4 2 w % w % q % a 4 3 p % v % r r % c = Z q q + Z w + X u r c = X q ? X u a 5 1 % w % w % a 5 3 q % u % r ? X w c Y v K p N r w = + % a 5 4 r % r r + % r % c = Y v ? Y p ? Y r c = X u + X w + X q c = X u + Z w + M q a 6 1 v % r p % r % a 6 2 u % r w % q % a 6 4 q % r q % q % c = Y v + K p + K r a 6 5 p % r p % r %

14. Nonlinear Low-frequency Vessel Model
Generalized damping and current forces: D Ý X Þ = D X + d Ý X , L Þ r L NL r r
where:

15. Examples of current coefficients surge, sway and yaw for supply ship:

16. Examples of current coefficients heave, roll and pitch for supply ship:

17. Damping properties

18. Nonlinear Low-frequency Vessel Model
Generalized restoring forces: G Ý R Þ

19. Nonlinear Low-frequency Vessel Model
Wind load :

20. Examples of wind coefficients surge, sway and yaw for supply ship:

21. Examples of wind coefficients heave, roll and pitch for supply ship:

22. Nonlinear Low-frequency Vessel Model
2. Order Wave loads :

23. Mooring System
Overview

24. Mooring System Single Line Modelling 3 types of excitation:
Large amplitude LF motions
Medium amplitude WF motions
Very high frequency vortex-induced
vibrations

25. Mooring System T H Line Characteristics 7000 5000 3000 1000
1370
1380
1390
1400
1410
1420
1430
Horizontal distance to anchor [m]

26. Mooring System Forces and moment on moored structure
Additional damping term
Restoring term

27. Mooring System Quasi-static mooring model:
Generalized mooring forces in LF model
Quasi-static mooring model:
- use the line characteristics
for each line i in

28. Mooring System
Linearized Mooring Model

29. Linear Wave-frequency Vessel Model
Potential 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).

30. Linear Wave-frequency Vessel Model
Linear 6 DOF Wave-frequency model - surge, sway, heave, roll, pitch and yaw :
Motion vector in hydrodynamic frame:
Earth-fixed motion vector:
1. Order wave loads

31. Verification tests on Varg FPSO

32. Full-scale results, Varg FPSO

33. Full-scale results, Varg FPSO