Motions of Moored Ships in Six Degrees of Freedom 



three components of translational motion of the origin of the moving 

 system relative to the fixed system are defined as the surge, the 

 sway, and the heave of the ship. The positive directions of forces and 

 moments are defined in the same way as their corresponding displa- 

 cements. 



The forces and moments on a moored ship can be divided 

 into four categories : inertia, damping, restoring and exciting forces 

 and moments. The details of determining these forces and moments 

 are discussed in [_5j , and only a brief discussion will be given be- 

 low. 



The inertia forces and moments arise from change of velo- 

 cities of the ship and water particles around it. Although the change 

 of velocity of water particles depends on their position in relation to 

 the ship, it can be assumed that a certain amount of water behaves 

 as if integral with the ship and moves with it. The amount of entrain- 

 ed water is different for different components of motion and the mass 

 or the moment of inertia of such entrained water is called the added 

 mass or added moment of inertia. By applying the slender body theo- 

 ry, the inertia force or moment on a section of the ship is then equal 

 to the product of the virtual mass (the sum of natural mass and added 

 mass) or the virtual moment of inertia (the sum of natural moment of 

 inertia and added moment of inertia) and acceleration. The added 

 mass and added moment of inertia depend on wave frequency, water 

 depth, shape of ship sections and the clearance between ship sections 

 and side walls. 



Damping forces and moments arise from wave generation 

 and are proportional to the relative velocity between the ship and wa- 

 ter particles. The proportional constants are called the damping coef- 

 ficients and depend on wave frequency, water depth, shape of ship 

 sections and the clearance between ship sections and side walls. 



Restoring forces and moments come from three different 

 origins and will be discussed separately in the following. 



Hydrostatic restoring forces and moments are due to the 

 buoyancy effect resulting from ship displacement. The total hydro- 

 static restoring force has only one component in the vertical direc- 

 tion and the hydrostatic restoring moments has components in the roll 

 and pitch direction. 



The second restoring forces and moments come from moo- 

 ring lines. The behavior of a mooring line under tension has been 

 investigated by Wilson [2] . The relationship between force and 

 elongation is highly nonlinear and in general can be represented in 



673 



