Evaluation of Motions of Marine Craft in Irregular Seas 



0(t) = 



K^(T) -n (t-T)dr 



(71) 



where k^(t) represents the so-called "system" impulsive response function 

 which combines the vertical shift in wave axis system [Eq. (59)] with the trans- 

 fer function of the submerged body in roll [Eq. (69)]. In effect, this system re- 

 sponse function directly relates the surface wave profile, t7o, with the motion of 

 the submerged body. This is the impulsive response function determined ex- 

 perimentally by Dalzell and reproduced in Fig. 8 of this report. The apparent 

 period of oscillation in Fig. 8 was equal to the natural roll frequency of the sub- 

 merged body which is as it should be. The logarithmic decrement of the oscil- 

 lation of the roll impulsive response function was calculated and found to agree 

 closely with the logarithmic decrement found from experimental roll decay 

 curves. The reason for the existence of K(t) for negative time (as shown in 

 Fig. 8) is that in the subject experiment the only available input time history 

 was that of the surface wave elevation directly over the body. Dalzell indicated 

 that the use of either the wave slope at depth or the hydrodynamic rolling mo- 

 ment as inputs would have led to only phase lags in the system (rather than lead 

 and lag angles as given in Eq. (66) and hence result in a so-called physically 

 realizable impulsive response function where K( t ) - for t < . 



Using the convolution integral in Eq. (71), the time history of roll motions 

 was computed by Dalzell and the results are reproduced in Fig. 6 of this paper. 

 As in the case of heave, it is seen that there is excellent agreement between 

 computed and experimental time histories. 



Fig. 8 - Roll impulsive response for submerged body 

 (derived from transfer function in Fig. 7) 



487 



