Mooring and Positioning of Vehicles in a Seaway 



a direction of positive rotation about the 5c, y, and z axes, 

 respectively, (i.e. port upward, bow downward and bow portward). 

 The positive directions of the forces and moments acting on the 

 body are similarly defined. 



The force (or moment) acting on the body is composed of the 

 inertial force due to dynamic body motions (denoted as Fj ) , the 

 force due to damping (denoted as F(j) , the force due to hydrostatic 

 restoring action (denoted as F^) , the force due to the moorings 

 (denoted as Fm) , and the force due to waves (denoted as Fyy). The 

 equations of motion are then established as 



ms = F = Fi + F, + F, + F^ + F^ (1) 



for rectilinear motions (with s representing any rectilinear dis- 

 placement, and m the mass of the ship), with similar representa- 

 tions for the angular motions, A discussion of these different types 

 of forces is given below, together with some results obtained, for 

 purposes of illustration. 



The hydrodynamic forces and moments due to dynamic body 

 motions are of inertial nature, and do not contain any terms of dissl- 

 pative nature. The effect of the free surface is accounted for by 

 different frequency-dependent factors that modify the added masses 

 of each section. All couplings of inertial nature are exhibited in the 

 results of the analysis. In the case of dynamic body motions, the 

 simplified results of slender-body theory states that the local force 

 on any section is equal to the negative time rate of change of fluid 

 momentum [ 2] . For the vertical force (z-force), this is expressed 

 by 



■dT" " dt l^33^bl» ^^^ 



where A„ is the added mass of the cross-section and w^^ is the 

 body vertical velocity, given by 



^b = ^t ^^ " ^^^ " ^ " ^^' ^^^ 



In the above equations, the coordinate ^ is a "dummy" variable 

 along the longitudinal coordinate x (and coincident with it) , and the 

 time derivative D/Dt is just the partial derivative b/dt, since there 

 is no forward speed. The quantity Aj^ is the added mass of the 

 cross section, including free-surface effects, which is obtained from 

 the work of Grim [5] for the class of sections known as Lewis forms. 

 The total vertical inertial force is then found to be 



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