Surf-riding and oscillations of a ship in quartering waves

The behavior of a ship encountering large regular waves from astern at low frequency is the object of investigation, with a parallel study of surf-riding and periodic motion paterns. First, the theoretical analysis of surf-riding is extended from purely following to quartering seas. Steady-state continuation is used to identify all possible surf-riding states for one wavelength. Examination of stability indicates the existence of stable and unstable states and predicts a new type of oscillatory surf-riding. Global analysis is also applied to determine the areas of state space which lead to surf-riding for a given ship and wave conditions. In the case of overtaking waves, the large rudder-yaw-surge oscillations of the vessel are examined, showing the mechanism and conditions responsible for loss of controllability at certain vessel headings.List of symbols c wave celerity (m/s) - C(p) roll damping moment (Ntm) - g acceleration of gravity (m/s²) - GM metacentric height (m) - H wave height (m) - I_x,I_z roll and yaw ship moments of inertia (kg m²) - k wave number (m^–1) - K_H,K_W,K_R hull reaction, wave, rudder, and propeller - K_p forces in the roll direction (Ntm) - m ship mass (kg) - n propeller rate of rotation (rpm) - N_H,N_W,N_R hull reaction, wave, rudder, and propeller - N_P moments in the yaw direction (Ntm) - p roll angular velocity (rad/s) - r rate-of-turn (rad/s) - R(,x) restoring moment (Ntm) - Res(u) ship resistance (Nt) - t time (s) - u surge velocity (m/s) - U vessel speed (m/s) - v sway velocity (m/s) - W ship weight (Nt) - x longitudinal position of the ship measured from the wave system (m) - x_G,z_G longitudinal and vertical center of gravity (m) - x_S longitudinal position of a ship section (S), in the ship-fixed system (m) - X_H,X_W,X_R hull reaction, wave, rudder, and propeller - X_P forces in the surge direction (Nt) - y transverse position of the ship, measured from the wave system (m) - Y_H,Y_W,Y_R hull reaction, wave, rudder, and propeller - Y_p forces in the sway direction (Nt) - z_Y vertical position of the point of action of the lateral reaction force during turn (m) - z_W vertical position of the point of action of the lateral wave force (m)Greek symbols angle of drift (rad) - rudder angle (rad) - wavelength (m) - position of the ship in the earth-fixed system (m) - water density (kg/m³) - angle of heel (rad) - heading angle (rad) - _e frequency of encounter (rad/s)Hydrodynamic coefficients K roll added mass - N_v,N_r yaw acceleration coefficients - N_vN_rN_rrN_rrv,N_vvr yaw velocity coefficients K. Spyrou: Ship behavior in quartering waves - X_u surge acceleration coefficient - X_uX_vr surge velocity coefficients - Y_v,Y_r sway acceleration coefficients - Y_v,Y_r,Y_vv,Y_rr,Y_vr sway velocity coefficientsEuropean Union-nominated Fellow of the Science and Technology Agency of Japan, Visiting Researcher, National Research Institute of Fisheries Engineering of Japan