ASSUMPTIONS ON DYNAMIC EQUATION 



The dynamic equations require an inertial frame, while the ship 

 geometry is constant in the ship frame. The study of ship dynamics 

 requires two sets of reference frames. Two frames coincide at the 

 undisturbed conditions and the origin is taken at the ship center of 

 gravity. As (d/dt) = (d/dt)^ + "x^ and neglecting the 

 flexibility of the snip, tne equations of motion^^for translation can be 

 written as: 



F. = M(X. + X. w. - X, w, ) 

 1 1 J J k k' 



If all the terms in the parentheses are of order e, then one can simplify 

 the above equation and obtain an uncoupled linear equation as: 



F. = M X. 

 1 1 



This assumption should be valid outside the range of resonance or 

 large motion amplitude. This, in turn, implies that the solution will 

 not be accurate at or close to resonance. Similarly, for moment equi- 

 librium, one can write: 



M. = I. . a. 



1 IX 1 



Even though the exclusion of nonlinear terms cannot be easily justified, 

 they are adopted for simplicity and linearization required in the other 

 building blocks of the model. The major question is how to interpret a 

 large value of motion. The formulation adopted is the one due to Kaplan 

 (1970). 



HYDRODYNAMIC FORCES AND MOMENTS 



The force acting on the ship can be decomposed into: 



F = F + F + F 



pressure viscous restoring 



F should, in general, represent the effect of fluid flow 

 about ?£e^sKip. This force is usually computed for the ship in equilib- 

 rium and fixed, assuming that the ship is transparent to incoming waves 

 (i.e, "Froude-Krylov hirpothesis") . The integration of pressure is 

 usually done on the still-water-wetted surface or its projection. This 

 calculation avoids harmonics that one can obtain by integrating up to 

 the calculated wave height. Motion-induced pressure forces are repre- 

 sented under the group of forces added mass and damping coefficients. 

 In their usage it is possible to see terms like 



F = M"(u - x) 



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