z 



— 'V-.^. [61 



where E^ is a heaving force function which depends on the waterplane coefficient and on the 

 ratio of wave length to ship length, X/L. 



The exact functional relationship between heave parameters and hull particulars is 

 difficult to determine analytically, especially since little is known concerning the effects of 

 forward speed and wave reflections from the sides of the ship on the heave force function. 

 The empirical curves given in Figures 11 through 13 indicate, however, that for constant 

 Froude numbers and tuning factors, the heave parameters of various hull forms are related to 

 the mean beam-draft ratio. These results can perhaps be used as a first approximation for the 

 heave amplitudes of hull forms which have characteristics similar to those of the five models 

 used in this investigation. 



The curves of heave parameter versus mean beam-draft ratio permit some interesting 

 comparisons between specific pairs of models. For example, cargo Models 1 and 2 have almost 

 identical section shapes and natural periods, but Model 2 has a larger midship beam-draft ratio 

 B/H. The smaller heave amplitudes of Model 2, therefore, are primarily due to a relatively 

 larger B/H rather than to differences in section characteristics. On the other hand. Model 4 

 has the same B/H as Model 2, but the finer section shapes of Model 4 result in a larger mean 

 beam-draft ratio C^B/C^H. The smaller heave amplitudes of Model 4 relative to Model 2, 

 therefore, can be attributed primarily to Model 4's relatively finer sections. 



PITCH AMPLITUDES 



The pitch parameter can be written in a form similar to that used for the heave parameter: 



</- 



^0^^ m 



The term E, is a pitching moment function similar to the heaving force function E^, in 

 that E I also depends on the waterplane coefficient and the ratio X/L. The pitch magnification 

 factor is given by 



^^ = t(l-Aj)2+K^2 ^2j-V4 [8] 



For normal ships, the natural pitching perioa is approximately equal to the natural heaving 

 period. The tuning factor A , can therefore be taken as equal to A^, and Kreitner's heave 

 period approximation can be used for calculating the natural pitching period. 



15 



