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: Asgeir.Sorensen@ntnu.no

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

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

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 …..

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

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)

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:

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

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:

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

Nonlinear Low-frequency Vessel Model System inertia matrix:

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

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 %

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:

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

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

Damping properties

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

Nonlinear Low-frequency Vessel Model Wind load :

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

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

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

Mooring System Overview

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

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

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

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

Mooring System Linearized Mooring Model

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).

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

Verification tests on Varg FPSO

Full-scale results, Varg FPSO

Full-scale results, Varg FPSO