I-M-in Yanq 



surge motion |_1J , that is, the moored ship is considered as having 

 only one degree of freedom, and it is known that the force-elongation 

 relationship of a mooring line is highly nonlinear [2] • Kaplan and 

 Putz [3] , and Muga [4] have investigated moored structures in six 

 degrees of freedom. In their study, the force -elongation relationship 

 of mooring lines is assumed to be linear, thus the problem is linear 

 and the solution can be readily obtained. In this paper, we consider a 

 more general six degree of freedom problem. The mooring force is 

 a nonlinear function of elongation ; since a ship can not be symmetric- 

 ally moored, and fenders are only at one side of the ship, motions of 

 the ship are asymmetric. An approach has been developed to generate 

 an approximate steady- state solution to this nonlinear asymmetric 

 problem. 



FORMULATION 



The motion of a moored ship in waves is an oscillating system 

 with six degrees of freedom corresponding to surge, heave, sway, 

 roll, pitch, and yaw. The ship is considered as a rigid body and its 

 deformation is neglected. Usually the length of a ship is much longer 

 than its beam, and the slender body theory can be applied to find the 

 hydrodynamic properties of a moored ship. According to this theory, 

 for an elongated body where lateral dimensions are small compared 

 to its length, the flow field at any cross -section is independent of that 

 at any other sections. Hence, the flow field of an elongated body, like 

 a ship, is reduced to a two dimensional problem of its cross- sections. 

 The total hydrodynamical properties is found by integrating over the 

 length of the body. 



The six dynamic variables surge, sway, heave, roll, pitch, 

 and yaw of a ship are expressed in terms of two right-hand cartesian 

 coordinate systems. A moving systems which is fixed in the ship, and 

 a fixed system which is fixed in space. The moving system (y ,y ,y ) 

 has its origin at the center of gravity of the ship and its three axes 

 (y > Y-p > y ) coincident with the three principal axes of the ship. The 

 y -axis is positive toward the bow, the y -axis is positive to port 

 and the y_ -axis is positive upward. This system will move with the 

 ship and the angular displacements about the three axes are respecti- 

 vely, the roll, the pitch and the yaw of the ship. They are positive for 

 rotations about the positive directions of y , y and y in a counter- 

 clockwise direction. The fixed systems(x , x, x ) is chosen such that the 

 two coordinate systems are coincident when the ship is at rest. Then the 



* Numbers in brackets designate References at the end of the paper. 



672 



