Ochi 



In the phase-plane diagram shown in Fig. 3(b), slamming occurs whenever the 

 circle crosses the line DC. Thus, the probability of occurrence of slamming is 

 given by 



Prob {Slam} = Prob {r = H, f > f^} 



r' 



r* . r=H 



—r+—r 

 R' R 



(5) 



where 



H - draft at the ship bow, 

 f ^ = threshold relative velocity, 

 Rr = twice the variance of relative motion, 

 r; = twice the variance of relative velocity = r' a; 2. 



r J" r o 



As can be seen in Eq. (5), it is necessary to evaluate the variances of rela- 

 tive motion and velocity for estimation of the probability. The application of the 

 superposition principle by using the response amplitude operators may be valid 

 to evaluate the variances even for conditions severe enough to induce slamming. 

 The justification of this statement will be given in the next section in which a 

 comparison between the predicted and measured probability of occurrence of 

 slamming are shown. 



The variances of relative motion and velocity at an arbitrary point along 

 the ship length can be approximately estimated from irregular wave tests also. 

 The method for evaluating the variances for this case is discussed in Appendix 1. 



Number of Slams per Unit Time 



The number of slamming occurrences per unit time is essentially an appli- 

 cation of the problem of the expected number of zero crossings per unit time. 

 The theory on the zero-crossing problem was first developed by Rice [6], and 

 later applied to ship slamming by Tick [1]. Therefore, the development of the 



552 



