Vassilopoulos and Mandel 



where the zero subscript denotes the dynamic equilibrium condition and, for 

 reasons to be subsequently discussed, the wave action forces and moments have 

 been lumped conveniently in Zg(h,h,h, t) and Mg(h,h,h, t) . Such a linearization 

 indicates that the forces and moments acting on a pitching and heaving ship may 

 be conveniently considered to be of two sorts: 



a. Wave-induced forces and moments acting on a restrained ship, and 



b. Forces and moments brought about by the motion of the ship in calm 

 water. 



Noting that z^ 



v^ = w^ = q^ = q^ = and using the notation 



3Z\ 



etc., Eqs. (21) and (22) become. 



Z^(h,h,h,t) + Z^ z„ + Z. ^ + Z^w + Z^q + Z. w + Z. q , 



(23) 



Mg(h,h,h, t) + M^ z^+Mg^+M^w+Mq + M.w+M.q. 



(24) 



Since it is desirable to express the differential equations in terms of the orien- 

 tation parameters z^ and d and their first and second time derivatives, an ex- 

 pression must be found for w and w in terms of z^ and e. From the following 

 sketch, 



it follows that. 



w = z„ COS t^ + u„ sin 



(25) 



or since within linearity, cos 



1 and sin (9 = e , we get 

 330 



