108 



THEORY OF SEAKEEPING 



face. The needed mathematical expressions reduce to 

 tractable forms only for deeply submerged ellipsoids. 

 Since the forces in this case are inertial, they can be ex- 

 pressed in terms of the "coefficients of accession to in- 

 ertia" k, defined as 



_ total inertia of a body floating in a fluid , ,- 

 inertia of fluid displaced by body 



In connection with the objectives of the present mono- 

 graph, interest is concentrated on prolate ellipsoids in 

 which the major semi-axis a is taken to coincide with the 

 .(■-axis in which also the mean body-velocity ^'ector V 

 lies. The minor semi-axes b and c (not necessaril.y ecjual ) 

 are then taken to coincide ^^^th the y and s-axes. The 

 oscillatory motion of the body may include translations 

 along any of the three axes and rotations about any of 

 these axes. The coefficients of accession to inertia have 

 different values for an}' of those motions, and the symbol 

 k is supplemented by a suitable subscript. Treating 

 the motions of an ellipsoid of revolution (a spheroid), 

 Lamb (Chapter 1-D) designated by ki the coefficient of 

 acces.sion to inertia for accelerations along the major 

 semi-axis a (i.e., .v-direction), by fe that for accelerations 

 along a minor axis, and by k' that for rotation about a 

 minor axis. These designations were used (among 

 others) by Davidson and Schiff (1946), Korvin-Krouko^■- 

 sky and Jacobs (1957, also Appendix C to this mono- 

 graph) and Alacagno and Landweber (1958). It has 

 been recommended' that symbols Av, k^, and k, be used 

 for translations along axes and /i'„, /.'„„, and k^. for ro- 

 tation about axes indicated by subscripts. This notation 

 was used by Weinblum and St . Denis ( 1 950 ) . ^\^lile con- 

 \'enient for treatment of multicomponent motion of three- 

 dimensional bodies this notation may be confusing in 

 discussing two-dimensional flows in the strip theory of 

 slender bodies. In this case it is customary to take the 

 .r-axis laterally in the plane of water and the ?/-axis ver- 

 tically. In order to avoid confusion with the three- 

 dimensional analysis the notation k„ and k^ can be used- 

 for the coefficients of accession to inertia in the vertical 

 and horrizontal (lateral) directions. Here A-, is identical 

 with k-i of Appendix C. 



A brief table of coefficients for spheroids will be found 

 in Lamb (Chapter 1, D, p. 155). Curves of the coeffi- 

 cients of accession to inertia or to moment of inertia for 

 various proportions of the ellipsoid axes can be found in 

 Zahm (1929), Kochin, Kibel and Rose (Chapter 1, C, pp. 

 385-389), and Weinblum and St. Denis (1950). 



Since the exact evaluation of the coefficients of acces- 

 sion to inertia of ship forms is practically impossible, it 

 has been customary to estimate them b.y comparison 

 with ellipsoids of similar length, beam and draft. A 

 tj'pical application of this method is found in the work 

 of Weinblum and St. Denis. 



' Minutes of the first meeting of the Nomenclature Task Group 

 of the Seakeeping Panel, SXAME. 



■ Subscripts r and h were used bv Landwelier and de Macagno 

 (1957). 



In making these estimates for surface ships an assump- 

 tion is introduced that the coefficients of accession to 

 inertia, initially derived for a deeply submerged body, are 

 still valid for a body floating on the surface. In other 

 words, the effects of wa\'emaking on the free water sur- 

 face are neglected. These effects have been investigated 

 in the simpler "strip theory" to be discussed in the next 

 section. It appears that, within the practical frecjuency 

 range, the coefficient k, for heaving oscillations of a float- 

 ing body may be, on the a\erage, SO per cent of that 

 computed by comparison with a deeply submerged ellip- 

 soid. 



It is clear that comparisons with ellipsoids are limited 

 to investigations of ship motions of a general nature, in 

 which only the over-all proportions are in\-olved and the 

 details of the hull form are not considered. In addi- 

 tion, the results are evidently applicable to investigation 

 of the motion of a ship, but provide no information on 

 distribution of forces along the length of a ship. Knowl- 

 edge of this distribution is neces.sary in calculating the 

 bending moments acting on a ship in waves. 



Theories and computations made for ellipsoids have 

 been important in bringing out certain trends or laws of 

 action of h.ydrodynamic forces which are indicative of 

 what can be expected in ships and submarines. As 

 typical examples of this theoretical activity, the work of 

 Ha\-elock (1954. 1955, 1956) and Wigley (1953) can be 

 cited. 



3 Evaluation of Forces in Heaving and Pitching by Strip Tlieory 



As has been mentioned earlier, solutions of three- 

 dimensional hydrodynamic problems ha\'e been limited 

 to ellipsoids, and are practically impossible when dealing 

 with ships.' The strip theory has been introduced in 

 order to replace a three-dimensional hydrodynamic prob- 

 lem by a sinnmation of t«'0-dimensional ones. Losing 

 this method, solutions are possible for a much wider range 

 of problems and actual hydrodynamic conditions con- 

 nected with ship motions can be represented more com- 

 pletely. F. AI. Lewis (1929) appears to be the first to 

 apply this theory in connection with e\-aluation of hydro- 

 dynamic forces acting on a vibrating ship. Hazen and 

 Nims (1940), St. Denis (1951), and St. Denis and Pierson 

 (1953) used the strip theory in connection with the analy- 

 sis of ship motions. This theory was described more 

 explicitly later by Korvin-Krouko\'sky (1955c) and 

 Korvin-Kroukovsky and Jacobs (1957). Quoting from 

 the latter work: 



"Consider a ship moving with a constant forward 

 velocity (T') (i.e., neglecting surging motion) with a 

 train of regular wa\'es of celerity (c). Assume the set of 

 co-ordinate axes fixed in the undisturbed water surface, 

 with the origin instantaneously located at the wave nodal 

 point preceding the wave rise, as shown in Fig. 1 [here- 

 with]. With increase in time t the axes remain fixed in 

 space, so that the water surface rises and falls in relation 



' Solutions of h.ydrodynamic problems related to special mathe- 

 maticall}' defined ship forms will be discussed in Section 6. 



