204 



I1M)R()1)\ .\ \Mlc:s i.\ Mill' DISIGN 



Sec. 19.16 



mntlipmaties, l\v tlic loiitniir or pmCilc dniwiiig 

 of the liuU ill tlic i)hvnc of symuielry. This in- 

 volves (usually) a straight kcol, a stem profile of 

 the selccteil tyjjc, and a stern profile to suit the 

 propeller anil nulder. 



Restrieting the calculations to these four lines 

 is on the basis that if four sets of points are estab- 

 lished on selectetl frames in accordance with the 

 foregoing, they should be sufficient to insure 

 that the principal dimensions and shape, as laid 

 ilown in any mold loft, will be as contemijlated 

 by the designer. Fig. •U).K illustrates the location 

 of these four sets of jjoints on a schematic body 

 plan. Calculation of offsets for shii)s of special 

 form may be made, of coui-se, for as many arldi- 

 tional line.'? as are desired. 



49.16 The Geometric Variation of Ship Forms. 

 Somewhat related to the use of mathematical 

 equations for ship lines is a combination of 

 mathematics and geometry for effecting a variation 

 of hull parameters and coefficients in a given ship 

 form. Perhaps the simplest procedure of this 

 kind is that in which the fore-and-aft position of 

 the maximum-area section is shifted along the 

 i-axis. Here, by holding the .same .sections and 

 section areas, the stations in the entrance are 

 closed up and those in the run are opened out, 

 or vice versa. Or, by closing up the sections in 

 both forebody and afterbody, a portion of parallel 

 middlcbodj' is inserted between them. The ship 

 length remains the same in both ca.ses. 



A variation is that in which a .ship is lengthened 

 by the addition of a middlebody. Usually the 

 cut is made at the maximum-scction-arca position 

 but this is by no means mandatory. If tlie length- 

 ening is accompanied by a repowering and a 

 speed increase, the new portion inserted may not 

 be of uniform section. It may well have a maxi- 

 mum-.scction area Ax greater than that of the 

 original ship. 



Geometric variations are almost unavoiduljle 

 when laying out model series in which one param- 

 eter is systematically varied, described by D. W. 

 Taylor [S and P, 1943, p. 55) and others. More 

 elaborate procedures arc explained and illustrated 

 in detail by II. Lackenl)y in his paper "On the 

 Systematic Geometrical Variation of Ship Forms" 

 (INA, Jul 19r>0, pp. 280-310]. 



When a geometric-variation jjrocedurc is applied 

 to a given ship form to produce; a new one there is 

 seriouH question whether the water will t!ik(; 

 kindly to the change. In other words, what 

 appoarH to be a good geometric or iiarametric 



variation may not work out so well hydnidynami- 

 cally. This situation wjus jjointed out by J. L. Kent 

 in his discu.s.sion of the Lackcnby paper [p. 310]. 



One may visualize a ship form of low total 

 resistance for its displacement, tiespite an LMA 

 position that appears by Fig. GG.L to be too far 

 forward for its speed-length (juotient and pris- 

 matic coefficient. Stretching the entrance and 

 contracting the run may be an easy way to 

 achieve what appears to be a better LMA po.sition 

 and what should give a slight but definite im- 

 provement in performance. There is no a.'vsurance, 

 however, in the present state of the art, that 

 becau.se of the geometric variation alone the 

 modified section-area curve, the modified designed 

 watcrlinc, and other altered features will produce 

 better hydrodynamic behavior. 



49.17 Selected References Relating to Mathe- 

 matical Lines for Ships, ("ortaiii rcfi'iciices which 

 describe methods developed in the past for 

 delineating the forms of bodies and ships are 

 listed here for convenience. They begin witli the 

 work of F. H. dc Chapman in the ITfiO's, although 

 his elTorts may not have been the first along 

 these lines: 



(1) Chapm.iii, F. II. dc, ".\ Trcati.se on Sliip-Builciiiig," 



translateti into English by the Rev. James Tnman, 

 Cambridge and London, 1S20 



(2) Nystrom, J. W., Jour. Franklin Inst., Jul-Dec 1S63, 



Third Series, Vol. XLVI, pp. 355-359 and 389-396, 

 with Pis. II and III 



(3) Jour. Franklin Inst., Jan-Jun 1864, Vol. XLVII, 



Third Series, pp. 4(>-5I, accompanied by Plate I 

 of the Nystrom scries of papers; also pp. 2-11-244 

 in the same volume 



(4) Jour. Franklin Inst., Jul-Dec 1864, Vol. XLVIII, 



Third Scries, pp. 261-261. This paper contains 

 some historical data on Cha|)inan'3 previous work 

 on mathematical lines (principally parabolas) in 

 Sweden. Plate IV of the series, mentioned here, i» 

 bound out of place (opp. p. 236) in the volume 

 consulted. 



(5) Taylor, D. W., "On Ships' Forms Derived by Form- 



ulae," SNAME, 1903, pp. 243-267 



(6) Warrington, J. N., "System of Mathematical Lines 



for Ships," SNAME, 1909, pp. ■Ml-1.')2 



(7) Taylor, D. W., "Calculations for Ships' Forms and 



the Light Thrown by Model Experiments upon 

 Resistance, Propulsion and Rolling of Shi[xs," 

 Trans. Int. Eng'g. Cong., lOl."), Naval .\rrhitecturc 

 and Marine Engineering, San Franci.sco, 1915 



(8) VVcinblum, G., "Beitriige zur Thcorio dcr Schiff- 



soberfl.iche (Contribution to the Theory of the 

 Ship Boundary)," WRH, 22 Nov l'.r>9, pp. 462- 

 461); 7 Dec 1929, pp. 489-493; 7 Jan 1930, pp. 12-14. 

 This paper illustrates a number of .ship forms whoso 

 lines were d('rive<l mathematically. 



(9) Woinblum, ('•. P., "ICxakt<; Wiuwerlinion und Spant- 



