In what follows cos X will be put equal to — 1, since the essential procedures 

 can be demonstrated by considering the ship heading directly into waves. 



Following traditions in naval architecture wide use use can be made of graphical 

 representation. 



As first item we consider the tuning factor. Although the discussion of the 

 latter represents a large part of the "theory" of roll, quite a bit remains to be done 

 in the present case. There is occasionally some confusion about the supercritical range 

 of ship operation due to the fact that for a given T the tuning factor can become 

 greater than unity either because 



COo/f > 1 



or because (23) 



1 + - ( = 1 + V ) is large. 



Obviously, when advancing in short waves the supercritical condition is the 

 natural one for larger ships. This case is, however, in general not interesting because 

 of the insignificant effects involved. The supercritical state becomes a problem in long 

 waves only. 



To answer our problems we need : 



1. The equation of the hull y(x, z), L, B, H, and the mass distribution. Call 



y* - y/L. 



2. Natural periods T z , T^ or frequencies. For simplicity we neglect here the 

 dependence of T upon w * (and a fortiori upon F), so that T = const. However, when 

 exploring new conditions (high speeds) it may become necessary to consider F(o)*) 

 as variable. In any case reasonable data are available to calculate J by a strip method 

 and applying corrections. 



An experimental determination of period and damping is recommended in 

 model research as a necessary prerequisite, notwithstanding the difficulty due to the 

 evaluation of pertinent extinction curves. 



3. Curves of damping in heave and pitch. 



We can assume k = k(m*, y*) neglecting the dependency upon F. K is best 

 calculated by a strip method introducing corrections when w * is small. As in the case 

 of period calculations the appropriate frequency parameter in general is u> B * — a>^/B/g. 

 The conversion to w * = o>yJL/g requires care. For the present purpose we use curves 

 in Fig. 1 and some averaged results. It appears fortunate that in the range of resonance 

 w* — v* = 2.5 — » 5 (5) we are not too far from the peak of the damping curve when 

 the lower values of v* apply. This may be the case, for example, when large mass 

 moments of inertia (in the supercritical range) are admitted. For large v * (v B *) the 

 decline in damping can become noteworthy. 



To avoid confusion the dimensionless coefficient « should be used for discus- 

 sions of this kind. 



The interval in which damping represents a decisive factor, say 



0.6 < A < 1.2 1.5 < a)* < 6 0.5 < Wii * < 2 



is large but nonetheless not excessive. There has been, naturally, a concern about a 

 possible loss of damping power when one departs strongly from present practice. It 

 turns out now that this loss can be safely estimated and does not become prohibitive 

 when v* is properly chosen. 



Particularly, the range of frequencies where three-dimensional effects are 

 pronounced, o* <^ 2.5 for heave and w * ^ 3 for pitch, lies normally at the border 

 or outside of the synchronism zone. The most important task at present appears to be 



83 



