sions are given for the lateral and rotational damping caused by such bilge (or center) 

 keels, showing that this damping increases with the fourth power of the keel height. 



The results for the ship without bilge keels are checked by another method 

 based on a conformal transformation. 



The same method is then applied to determine the heave motion in calm water. 

 Expressions are obtained for the added mass which in principle agree with Haskind's 

 findings; the known result for the damping is restated. 



A bold attempt is further made to compute the damping (for example for an 

 elliptic section) when the rolling angle is no more considered as small. 



Boundary conditions are now fulfilled at the actual condition and not at the 

 state of rest. The magnitude of the damping coefficient obtained in this way may be 

 appreciably higher than by the standard approach. 



The exciting hydrodynamic forces caused by regular waves are determined 

 by assuming first that the body is fixed. When the body is released these generalized 

 exciting forces determine orbital motions of the body. 



By considering quadratic terms in — in the development of the wave potential, 



g 

 further results are obtained. It can be shown within the range of validity of the develop- 

 ment made that the heaving motion does not exceed appreciably the orbital motion 

 even at synchronism. This statement, which represents an extension of some results 

 due to Ursell, settles the question raised by Chadwick [4]. 



Expressions for the exciting moment of roll are extremely interesting. They 

 depend appreciably upon the shape of the cross section. It is imaginable, at least in 

 principle, to design sections which experience low exciting moment at synchronism. 



Unfortunately, it was impossible so far to establish agreement between the 

 fundamental equations of forced roll used by Woznessensky [9] and by Grim. A lot 

 remains to be done to reach a satisfactory state of knowledge in this seemingly outworn 

 field of research, and results of high interest from the viewpoint of theory as well as 

 practice can be expected from further work. 



3.6. Coupling of Motions and Equations of Motion 



This problem has been discussed already at some length at the VII International 

 Tank conference. In the meanwhile important contributions have been made, to some 

 extent inspired by the analogy with flight mechanics, but otherwise following lines 

 established primarily by Haskind. Havelock [46] has recently reconsidered the problem 

 of coupled free heave and pitch treated by Haskind, neglecting damping of the motions. 

 He shows that the influence of coupling on natural periods is negligible. 



The interesting technical problem which involves the discussions of forced 

 oscillations so far remains open. It was raised by Grim's experiments showing large 

 heaving effects due to forced pitching. 



Korvin-Kroukovsky has tried to give a plausible physical interpretation of 

 coupling terms in the combined heave and pitch equations 



m'z Z z + N 3S z + cz + ra 35 <A + N z 4 + # = Fe imt (12) 



m'*$ + Ns4 + C$ + m 35 z + N 35 z + Gz = Me iat (13) 



From his general analysis of forces he obtains 



m 35 — m 33 , JV 35 = N 53 . 



The coefficients are calculated from added mass and damping values in heave per unit 

 length of the ship by integration. It is further shown that C and g should be split up 

 into C zz C\ + C 2 and g zz g 1 + g 2 where C t , g 1 are due to hydrostatic and C 2 , g z 

 to hydrodynamic effects. These latter effects have not been considered by Haskind. 

 Prof. Korvin emphasizes some shortcomings in his analysis which can be eliminated 



74 



