Ship Maneuvering in Deep and Confined Waters 



to the right of the bar may be dropped. Terms containing the coef- 

 ficient Yf. have been retained in view of the fore-and-aft unsymmetry 

 present particularly in propelled bodies. 



The coefficients for u in X, for v in Y, and for r in N 

 — with signs reversed — are the most commonly well-known added 

 masses and added moment of inertia respectively. These inertia 

 coefficients also appear in some of the cross -coupling terms. 



Lamb's "coefficients of accession to inertia" relate added 

 masses to the mass of the displaced volume V (kjj , i = 1, 2, 3) and 

 added moments of inertia to the proper moments of inertia of the 

 same displaced volume (k-j , i = 4, 5, 6), Lamb calculated k||, 

 ^22~ ^33 ^^^ ^55~ ^66 ^^^^ ^^^ sphereoid of any length-to-diameter 

 ratio, [15], For ellipsoids with three unequal axes the six different 

 coefficients were derived by Gurewitsch and Riemann; convenient 

 graphs are included in Ref. [ 16] , For elongated bodies in general 

 the total added inertias may be calculated from knowledge of two- 

 dimensional section values by strip methods, applying the concept 

 of an equivalent ellipsoid in correcting for three-dimensional end 

 effects, (See further below.) 



Of special interest in Eq, (5.6) is the coefficient Y^ - X^ 

 in the "Munk moment," [ 17], (See also discussion in [ 18] ,) This 

 free broaching moment in the stationary oblique translation within 

 an ideal fluid defines the derivatives 



L^ 



n;v=- T ' %=^<^^^-^^^) (5-7) 



(Cf, Table II,) The factor kgg- k|| may be looked upon as a three- 

 dimensional correction factor. 



Due to energy losses in the viscous flow of a real fluid past 

 a submerged body the potential flow picture breaks down in the 

 afterbody. In oblique motion there appears a stabilizing viscous 

 side force. So far no theory is available for the calculation of this 

 force, but semi-empirical formulas give reasonable results for con- 

 ventional bodies of revolution. Force measurements on a divided 

 double-body model of a cargo ship form have demonstrated that some 

 de- stabilizing force is still carried on the afterbody but that most of 

 the moment is due to the side force on the forebody, predictable 

 from low-aspect-ratio wing or slender body theories , [ 18], 



Similar measurements on a divided body in a rotating arm 

 shall be encouraged. Contrary to the case of stationary pure trans- 

 lation the pure rotation in an ideal fluid involves non-zero axial and 

 lateral forces. From Eq, (5,6) the side force is given by X^ur, 

 whereas the moment here is Y^ur. For bodies of revolution the 

 distribution of the lateral force may be calculated as shown by Munk 



831 



