Sec. 49.5 



MATHEMATICAL LINES FOR SHIPS 



191 



origin at midlength is expressed as the linear 

 distance Xb or as the 0-diml ratio Xb/{L/2), plus 

 if forward and minus aft. 



49.5 Application of the Dimensionless Surface 

 Equation to Ship-Shaped Forms. The 0-diml 

 shape or surface function ■t]{^, f) of the hull may 

 take a great variety of forms, even for boat- or 

 ship-shaped underwater bodies. The simplest are 

 the binomial forms 



nik, 0) = 1 - r 



'7(0, f) = 1 - r" 



(49.vii) 



(49.viii) 



These produce what are called Chapman parab- 

 olas [Weinblum, G. P., TMB Rep. 886, pp. 13- 

 14]. With different numerical values of the 

 exponent n other than 1.0, but not necessarily 

 whole numbers, the shape function of Eq. (49.vii) 

 gives parabolic waterlines of varying curvature 

 and fullness, much like those described and dis- 

 cussed by J. W. Nystrom in the 1863 and 1864 

 Franklin Institute references quoted in Sees. 

 49.3 and 49.17. 



The shape selected for the designed waterline, 

 in the simplest case, determmes the function of the 

 0-diml fore-and-aft distance ^ from the origin. 

 The shape selected for the midsection determines 

 the function of 0-diml keelward distance f from 

 the origin. Thus the shape defined by the ex- 

 pression 7/ = 1 — ^" of Eq. (49.vii) has an nth- 

 order parabolic waterline and wall sides. That 

 defined by t; = 1 — f "" has mth-order parabolic 

 sections and square (plumb) ends. 



It is possible to shape the waterlines and sec- 

 tions independently by using a surface equation 

 in the form of the binomial product 



vii, f) = (1 - r)(i - n (49.ix) 



A half-body plan of a hull developed by putting 

 n = 2 and m = 2 is published by Weinblum 

 [TMB Rep. 886, Fig. 3, p. 20]. 



Introducing what he calls a fining function 

 into the first binomial of the product [TMB Rep. 

 886, p. 21], Weinblum produces a 0-diml equation 

 of the form 



'!(?, f) = [1 - r 



— (a coefficient) (^" 



f)f](l - n (49.x) 



„ = [1 - f - (0.5757)(f - ^^)f](l - f) (49.xa) 



With this surface function Weinblum produces 

 the half-body plan of Fig. 49. B, adapted from 

 Fig. 4 on page 21 of TMB Report 886. This has a 

 marked resemblance to some actual ship forms. 



When given a specific set of values, for example 

 by using a selected coefiicient and putting n = 2, 

 m = 2, p = 4, and 2 = 9, Eq. (49.x) becomes 



Fig. 49.B Body Plan of a Schematic Ship with 

 Mathematical Sections Developed by Weinblum 



Evaluation of the 0-diml transverse offsets from 

 Eq. (49. xa) is fairly simple for a computer with a 

 desk calculating machine. For example, setting 

 the value of f as 0.4, for a section just abaft the 

 forward quarter point, and the value of f as 0.5, 

 for the half-draft waterline, Eq. (49.xa) becomes 



)j = {1 - (0.4)' - (0.5757) 



• [(0.4)' - (0.4)^](0.5) ! [1 - (0.5)'] (49.xb) 



The 0-diml value of 77 for the point marked with 

 a diagonal cross in Fig. 49.B is found to be 

 0.79975. For a value of i, equal to 0.5 and the same 

 0-diml waterline at f = 0.5, the 0-diml value of 7/ 

 for the point marked with a small circle is 0.6947. 

 If, in the hull-surface equation (49. xa), f is 

 put equal to zero, the result is a 0-diml equation 

 for the surface waterline, namely jj = (1 — ^ ). 

 If this equation is differentiated mth respect to f, 

 the d'i\ld^ equation resulting gives the 0-diml 

 slope of the surface waterline at any selected value 

 of ^. If a second differentiation with respect to 

 f is made, there is obtained an equation for 

 ^rildt which gives the rate of change of 0-diml 

 waterline slope for any value of %. These are 

 converted into ship values in terms of x and y 



