Understanding and Prediction of Ship Motions 



6 6 6 



6 ^t 



k= 1 -'- OD 



(T) K..(t-T)dT, (12) 



^jk 



where x^^ is a steady force component, /x-^ is a constant depending only on 

 ship geometry, bj^ and c^^ are constants which depend on ship geometry and 

 forward speed, and Kj^(t) is a function of time, geometry, and speed. None of 

 these quantities depends on the past history of the unsteady motion. X- repre- 

 sents the total hydrodynamic and hydrostatic force and moment on the ship due 

 to its own motions, plus the static buoyancy and drag forces on the ship in its 

 equilibrium position. In order to obtain the equations of motion, we must add to 

 X-(t) the other forces acting on the ship, viz., the wave-induced forces, body 

 (gravity) force, artificial restraints, propulsive force, etc., and set this sum 

 equal to the inertial reactions, in accordance with Newton's Law. 



First, we note that Xj^ will be exactly offset by the steady propulsive force 

 and by the gravity force on the ship, for we assume that the perturbations occur 

 in a system which is otherwise in equilibrium. Therefore we can omit both of 

 these external forces if we also set X^^ equal to zero. 



Let us denote by Fj(t) the six components of force and moment due to inci- 

 dent waves and by Gj(t) the six components of all other external forces and 

 moments (except the two steady components of force). Then the equations of 

 motion are: 



6 



E "^jk^kCt) = Xj(t) + Fj(t) + G.(t) - /3.mga.(t) , (13) 



where 



/3. = X3*, for k = 4, 5, 



= 0, otherwise; 



X3* = vertical distance of center of gravity below the origin of the coordi- 

 nate system in the equilibrium position, 



m.^ = generalized mass such that, if t = kinetic energy of the ship, 



6 



E" , X d 3t 



"jk-k(t) = d-t ^- 

 k= 1 k 



If the ship has lateral symmetry, it is readily found that: 



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