DERIVATION OF DISTRIBUTIONS OF SHIP MOTIONS AND LONGITUDINAL 

 BENDING MOMENTS OF THE HULL 



It will be assumed without further discussion that the short-term distribution of wave- 

 induced ship motions and stresses may be represented by the one-parameter Rayleigh distri- 

 bution and Uiat the corresponding long-term distributions are approximated by the two- 

 parameter log-normal distribution. Evidence to support these hypotheses is presented in Ref- 

 erence 3. 



Typical distribution patterns of variation* in pitch angle are shown in Figures 3 through 

 6. In all, 129 similar sets were analyzed. Pertinent results are given in Tables 2 through 6 

 for variations of pitch angle, pitch acceleration, roll angle, heave acceleration, and the hull 

 girder stress in the main deck amidships due to bending of the ship in a longitudinal plane 

 normal to the deck. 



It is interesting to note that all cumulative Rayleigh distributions (for example, those 

 shown in Figures 4 and 6) become identical if v"^ = x^fE is plotted against the probability 

 instead of plotting x directly. Utilizing this artifice it is necessary to know only the value of 

 E corresponding to a particular sea condition, ship speed, and heading in order to obtain the 

 probability of exceeding any value of x from a single graph (Figure 4) which is equally appli- 

 cable to wave heights, ship motions, and hull stresses. The values of E for various ship 

 operations are given in Tables 2 through 6. Table 7 gives factors which, togetiier with the 

 value £", permit making statistical predictions as discussed later. 



We now proceed to utilize the short>term distributions, each of which is characterized 

 by a value of £, as building blocks in order to construct the long-term frequency distribution 

 patterns of the ship responses to tJie sea applicable to wartime service in the North Atlantic 

 Ocean. (It should be noted that the distribution patterns for other "missions" can be readily 

 computed from the data given in this report.) Each of these short-term distributions will be 

 weighted in accordance with the relative fraction of time spent at given sea state (/,)» ^^ ^^ 

 given heading to the sea (/,), and at the given ship speed (/ ). For example, if tests have in- 

 dicated that the ship will experience N = 480 pitch variations per hour in a State 2 sea when 

 heading directly into the waves at a speed of 10 knots, then one may expect that n = f-Jofz^ 

 = (0,33) (0.34) (0.125) 480 = 6.73 variations of pitch angle per hour, out of the average 

 number of variations per hour, can be attributed to this set of environmental conditions over 

 an average year's operation in the assigned mission. 



These calculations are carried out in Tables 8 through 11. Each horizontal line in 

 these tables gives the data corresponding to a given set of environmental conditions. The 

 probabilities (1-P) of exceeding given values of pitch angle, etc., are computed and tabulated 

 in columns 10 through 18. The total number of variations per hour which, over the average 



*Throu^out this report, a variation is taken to mean the peak-to-peak Tailation of the vanablck 



7 



