Prediction of Ship Slamming at Sea 



The relative motion is considered as a random variable having a narrow- 

 band normal distribution with zero mean, since the relative motion is a combi- 

 nation of pitch, heave, and wave motions, all of which have narrow-band normal 

 distribution with zero mean. The relative motion is expressed by the following 

 formula 



r(t) = r„(t) cos {a;^t+ e^(t)} (1) 



where 



r^(t) = amplitude of the envelope of the relative motion, 

 oj^ = expected frequency = CT. /a J. , 

 e^ = slowly varying phase angle, 

 a^ = variance of relative motion, 

 0-.2 = variance of relative velocity. 



r 



It is noted that the relation co^ = a./a^ holds since a narrow-band normal 

 process with zero mean is considered. Assuming that f and k are small for 

 a narrow-band normal process, the following equation is derived from Eq. (1). 



r' = r^ + -^- . (2) 



Now, the probability density function of r^(t) is a Rayleigh distribution. 

 Since slamming occurs only when the relative motion is positive, the probability 

 density function of the positive r (t) can be written by 



f(r<.) - ,, 



o 



2r„ "rT 



(3) 



Note that Eq. (3) represents the probability density function of the cross 

 points on the OA-line in Fig. 3(b), and that the parameter, R^, involved in the 

 equation is not eight times but is twice the variance of the relative motion. 

 Hence R^ is equal to the cumulative energy density, i.e., the area under the en- 

 ergy spectrum, E, using the St. Denis- Pierson definition of the spectmm. From 

 Eqs. (2) and (3), 



f(rj = e \ ^ / . 



As was mentioned earlier, slamming occurs when the relative velocity ex- 

 ceeds the threshold velocity at the instant of reentry, i.e., r = h, and f > r^. 



551 



