Understanding and Prediction of Ship Motions 



In Eq. (8) the other important parameter for the waves, that is, the direction of 

 encounter, 6^^ is no longer in evidence. 



If o)^ and 6^ and the region of definition for that portion of the spectrum are 

 known then a particvilar wave, T;^j(t) can be associated with the motion of a ship 

 as caused by this one sinusoidal wave. The concept of a transfer function as 

 evaluated at a particular frequency for a particular direction is then valid and 

 the response of the ship can be considered as a part in phase with the forcing 

 waves or 180 degrees out of phase and a part in quadrature with the forcing 

 waves having a phase of either 90° or 270° with the forcing waves. The func- 

 tions, c^C^e' ^e^> QzC'^e'^e^j ^^^ ^° ^^» ^^'^ ^^ determined either by means of 

 experiments or by theories. The response to a single forcing wave is therefore 

 given by Eqs. (9), (10), and (11) for heave, pitch, and roll. 



Zj(t) = ac^Coj^p^^l) cos (oj^^t+e) + aq^Cw^p^^l) sin (oj^jt+e) (9) 

 0j(t) = ac^(a)^l,e^^) cos (oj^it+e) + aq^Co^^p ^^l) sin (oj^^t + e) (10) 

 0i(t) = ac^(ajgi,^gl) cos (w^^t + e) + aq^(aj^j , 0^l) sin (a;^^t+e). (11) 



The difference between short crested waves and long crested head seas is 

 most striking here. In general, if 6^ is changed, c^, q^, c_^, and q^ may or 

 may not change but they can change even for the same frequency of encounter. 

 For example, one wave at +30° into the course of the vessel and another at -30° 

 to the course of the vessel will look the same at the center of gravity of the 

 vessel, but they will produce rolling motions that are 180° out of phase with 

 each other. 



From Eqs. (9), (10), and (11) by means of the definition of the seaway of 

 encounter given by Eq. (5), it is possible now to write down the vector Gaussian 

 process that describes the time histories that would be recorded for the forcing 

 seaway (at the center of gravity of the vessel if it could be observed there) and 

 the heaving motion, the pitching motion, and the rolling motion. 



77g(t) = 12 cos (a,-^t + e) ^P^J^^^^^6^)^c^^^e^ 

 z(t) = 21 [cos(a)^t + e) c^(a;^,e^)+ sin (a;^t + e) Q^C^e' ^e^] V2Sg(a;^, 5^) Ao;^, A^ 



(12) 



V/(t) = 2l[cos(c^gt + e) c^(a)^,£?g) + sin (oj^t + e) q/^e-^e)] 4^^^^^^^^^^^'^)'^^^^^^ 



0(t) = I2[cos(a;^t+ e) C^(a;^,0^)+ sin (oj^t + e) q<^(^e' ^e)] \/2Se(^e' ^e^ ^'^e' ^^e • 



These double summation partial sums are to be evaluated over the same 

 net in w^ and 6^ for the same random phases. If there is any "phase relation- 

 ship" between the various motions and the forcing seaway these phase relation- 

 ships will be preserved. However, by virtue of the remarks made above in con- 

 nection with Eqs. (9), (10), and (11), it is not necessary for a particular phase 

 relationship to manifest itself. Indeed, in general, there are cases where no 



83 



