Nonlinear Time-Domain Strip Theory Formulation for Low-Speed Manoeuvering and Station-Keeping

Abstract

This paper presents a computer effective nonlinear time-domain strip theory formulation for dynamic positioning (DP) and low-speed manoeuvring. Strip theory or 2D potential theory, where the ship is divided in 20 to 30 cross sections, can be used to compute the potential coefficients (added mass and potential damping) and the exciting wave loads (Froude-Krylov and diffraction forces). Commercially available programs are ShipX (VERES) by Marintek (Fathi, 2004) and SEAWAY by Amarcon (Journee & Adegeest, 2003), for instance. The proposed method can easily be extended to utilize other strip theory formulations or 3-D potential programs like WAMIT (2004). The frequency dependent potential damping, which in classic theory results in a convolution integral not suited for real-time simulation, is compactly represented by using the state-space formulation of Kristiansen & Egeland (2003). The separation of the vessel model into a low-frequency model (represented by zero- frequency added mass and damping) and a wave-frequency model (represented by motion transfer functions or RAOs), which is commonly used for simulation, is hence made superfluous. Transformations of motions and coefficients between different coordinate systems and origins, i.e. data frame, hydrodynamic frame, body frame, inertial frame etc., are put into the rigid framework of Fossen (1994, 2002). The kinematic equations of motion are formulated in a compact nonlinear vector representation and the classical kinematic assumption that the Euler angles are small is removed. This is important for computation of accurate control forces at higher roll and pitch angles. The hydrodynamic forces in the steadily translating hydrodynamic reference frame (equilibrium axes) are, however, assumed tobe linear. Recipes for computation of retardation functions are presented and frequency dependent viscous damping is included. Emphasis is placed on numerical computations and representation of the data from VERES and SEAWAY in Matlab/Simulink. For this purpose a Simulink add-in to the Marine Systems Simulator (MSS) at the Norwegian University of Science and Technology has been developed (Fossen et al., 2004).