190 



HVDROnVNAMICS I\ SIIIT nisicx 



.S>r. -IP.-) 



ing somewhat to tlic uiulinvator IukIv of a Xortli 

 American Iiuliaii eaiioe. 



The general dimensional equation of the hull 

 surface is then 



±l(f. f) 



(49. iv) 



y = ±y(x, z) 



(49.i) 



where the plus and mmus signs represent identical 

 transverse offsets to starboard and port, respec- 

 tively, and the expression y{x, z) signifies a 

 transverse (/-function for the hull. In this case 

 it is a function of both x and z. For example, if 

 X were 21"j ft forward of the amidships origin on 

 a certain vessel, and z were 20 ft below the origin 

 (the latter in the iilane of the designed waterline), 

 y would be some value such as ±14.2 ft, measured 

 to starboard and to port, by virtue of the func- 

 tion ifc(/(.r, z). If the j/-function were 1.0 for 

 any combination of absolute values of x and z, 

 then the starboard and port «/-ofTsets would be 

 etjual throughout. When defined by suitable x- 

 and 2-limits the craft would have the form of a 

 wall-sidetl box or parallelepiped. 



If the craft were un.sjnnmetrical fore and aft 

 about the midlength it would be necessary 

 normally to use two surface equations: 



yy = zkyrix, z) for the forebodj' (49.iia) 

 y^ = ±2/4 (x, 2) for the afterbody (49.iib) 



This procedure has the effect, however, of 

 destroying the ship as an entit}-, with a single 

 origin and a single surface ec|uation. Carried to 

 its logical conclusion it would break up the ship 

 into two di.s.similar half-bodies, one comprising 

 the entrance and the other the run, each with its 

 own set of surface, area, volume, and moment 

 equations. To avoid losing the single ship equa- 

 tion, which may later be introduced into opera- 

 tions involving wavemaking, maneuvering, and 

 wavegoing, Weinblum has developed a special 

 proci'<lun' for asymmetry, to be dcscrilicd pres- 

 ently. 



Ueturning to the simple ship, .symmetrical 

 fore and aft, the next step is to convert the x-, y-, 

 and z-fifTscta and coordinates to 0-tliml fonn. 

 When this m done they become 



KkBi) - f , ij(eta) = | , r(7,eta) = j^ (lO.iii) 



2 2 



The (l-<linil form of tiie general (>(|ualion (l'.).i) 

 beconiex, adopting W'cinbluin's notation, 



where Tj((, f) is a transverse 0-diml j;-funclion of 

 the iiull. In this case it is a function of both f and f . 

 The general surface equations serve eciuallj' well 

 as equations of the usual ship lines when one of 

 the set of coordinates is given a fixed value. For 

 the surface waterline, where z or f is zero, the 

 ef mat ions become 



7/ = ±2/(.r, 0) and ,; = ±T,(f, 0) (49.v) 



For the niiflscction, whore .r and { are zero, 



y = ±2/(0, z) and r, = ±»,(0, f) (49.vi) 



For a submarine hull which is not a body of 

 revolution and is not sjTnmetrical above and 

 below any horizontal reference plane, there would 

 be required two sets of surface equations, one for 

 the upper and one for the lower portion. The 

 common origin for the two would lie in some 

 convenient horizontal axis or dividing plane. 

 Again, however, this would break up the vessel 

 as an entity by splitting it horizontally. A pro- 

 cedure would have to be devised which would 

 retain a single ship equation for use in analytic 

 studies, say of underwater maneuvering. 



Weinblum's general or ba.sic procedure for hulls 

 that are asymmetric with respect to the midlength 

 section is to split up the surface equation into a 

 main part that is symmetric fore and aft and an 

 "asymmetric (skew) deviation" represented by 

 a .secondary e(|uation [TMH Kep. SSG, .lun 1955, 

 p. 10]. Procedures and eciuations for the situation 

 where the section of maximum area is not at 

 midlength, with unequal entrance and run 

 lengths, anil for other asymmetric variations, 

 become somewhat involved. One such case is that 

 in which the maxinunn waterline beam does not 

 occur at midlength. The procedures involved are 

 not discu.ssed here but the reader who wishes to 

 study them may consult page 9 and following of 

 Weinblum's TMB Report SS(>, issued in June 

 1955, as well as his earlier paper [STCi, l'.)5l{, pj). 

 180-215]. 



Weinblum has pointed onl liial as more and 

 more mathematical lines ami niatiiematical ex- 

 pressions for ship form come into use, whether 

 analytical or general, it becomes necessjiry to use 

 ratios or new symbols for (juantities formerly 

 expressed by abbreviations (SNAME, 19IS, j). 

 4l.'i]. A ca.se in i)oiiil is the u.se of the ratio IX"H, 

 with a reference jioint at the FP. Using the ship 

 axes of I''ig. l.K, the distance of (he ( 'H from the 



