Sec. 62.2 



Form of 

 Three-Dimenaional Body 



ESTIMATE OF ADDED LIQUID MASS 



421 



Added Moss of 

 Entrained Liquid 



Sphere 



Circular Disc 

 Movinq Normal 



to Its Plane 



or Rototinq 



About Q Diametei 



Added Moment 

 of Inertia of 

 Entrained Liquid 



'^/OTrSL^ 



iL=-3-/)Tra m\_--j-jova. 



Mode of Motion 



cIS 



Mode of 

 Motion 



^L=z|/'a= 



Movi nq m l" ^z^^'"^^") 



Broadside 



Fig. 62.B Added-Liquid-Mass Valtjes for Some 



Thrbe-Dimensional Geometric Shapes in Unsteady 



Motion 



The respective modes of motion are indicated by the 



double-headed arrows. 



Washington, 1932. Formulas giving the inertia 

 coefficients of a variety of 2-diml and 3-diml 

 shapes developed from the general or elliptic 

 ellipsoid having semiaxes a, b, and c, and enabling 

 the added masses (and added mass moments of 

 inertia) of entrained liquid to be calculated for 

 translational motion along, and for rotational 

 motion about the three major axes, are given by 

 A. F. Zahm [NACA Rep. 323, 1929, Part V, 

 Table VIII, p. 445]. 



G. P. Weinblum and M. St. Denis present, in 

 graphic form, values of the three linear and the 

 three angular inertia coefficients for the elliptic 

 ellipsoid, again for translational motion along, or 

 for rotational motion about the three principal 

 axes. These data cover a range of ratios between 

 the semiaxes a, b, and c corresponding to the 

 proportions of normal ships [SNAME, 1950, 

 Figs. 6-11, pp. 189-190]. For use with added-mass 

 values for the general ellipsoid, the body mass 

 Wb of a buoyant ellipsoid in a liquid of mass 

 density p is (4/3)irpa6c. 



As set down in the SNAME paper and in the 

 list to follow, each of these linear (and angular) 

 inertia coefficients represents the ratio between 



(1) the added-liquid mass (or mass moment of 

 inertia) about the complete elliptic ellipsoid to 



(2) the mass (or the mass moment of inertia) of 

 the complete ellipsoid along or about the axis 

 specified, when it has the same mass density 

 as the displaced liquid. For a half-ellipsoid 

 representing the underwater body of a surface 

 ship, these added-mass (or added mass moment 

 of inertia) values are all halved but the ratios 

 and the coefficients remain the same. 



It is interesting to note that, in some cases, 

 the mass of half of an elliptic ellipsoid is re- 

 markably close to the mass of the water displaced 

 by the underwater hull of a ship of the same 

 principal dimensions. For example, in the case 

 of the ABC ship designed in Part 4, a half- 

 ellipsoid having the same proportions and size 

 as the ship has the following dimensions: 



Semimajor (longitudinal) axis a = L/2 = 

 510/2 = 255 ft, from Table 66.e. 



Seraiminor (transverse) axis b = Bx/2 = 

 73/2 = 36.5 ft 



Semiminor (vertical) axis c = H {not H/2) = 

 26 ft. 



The half -volume of this ellipsoid is (0.5) (4/3) 

 Tabc. The weight of the buoyant half-ellipsoid is, 

 in lb, the half-volume times p times g. Hence the 

 weight displacement of the half-ellipsoid in salt 

 water at sea level is 



l^(long tons) 



1 

 2 



4 \ -rrpgiabc) 

 3/ 2,240 



3.1416(1.9905)32.174(2.55)36.5(26) 

 2,240 



= 14,490 1. 



