Prediction of Ship Slamming at Sea 



Prediction of the Magnitude of Relative Velocity Between 

 Wave and Ship Bow 



In order to predict the magnitude of relative velocity between wave and the 

 ship bow, the probability density function of the relative velocity associated with 

 slamming must be established. In other words, the probability density function 

 of the cross points on the DC-line shown in Fig. 3(b) should be obtained. Al- 

 though the relative velocity associated with slamming was defined as negative, 

 the sign will be changed hereafter for convenience. 



Since slamming occurs when the relative motion is equal to H, let r = H in 

 Eq. (2). Then, 



r 2 = h2 + -^ . (8) 



Consider the probability density function of r^ when r^ is greater than h; 

 namely, consider the probability density function of the cross points on the BA- 

 line shown in Fig. 3(b). The result is 



From Eqs. (8) and (9), 





ir(rln^) (9) 



2 



K . (10) 



f(r) = 1^ e ' , f > 



r 



Thus, the probability density function of the cross points on the BC-line in 

 Fig. 3(b), neglecting the sign of the relative velocity, is the Rayleigh distribution. 



Next consider the probability density function of the cross points on the 

 DC-line in the figure, since it is necessary to consider the threshold velocity, 

 r^, for slamming. Then, the probability density function of the relative velocity 

 for slamming is given by 



^^'^ - r: 



ir(r'-rl) (11) 



where 



r: = twice the variance of relative velocity, 



r 



r, = threshold velocity. 



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