Sec. 49.13 



MATHEMATICAL IJNES FOR ,S?IIP.S 



199 



approximate very closely the A-curves of actual 

 ships and models [SNAME, 1953, pp. 580-582]. 

 Having the equations of these curves, a ftirther 

 development along the lines proposed by R. 

 Taggart, using methods similar to those described 

 in Sees. 49.13 and 49.14, should serve for the 

 mathematic fairing of the curves in question. 



As a check on the fairing of any section-area 

 curve, whether represented by a mathematical 

 equation or not, the dimensionless curvature 

 may be determined as described for a waterline 

 in Sec. 49.10. For the A/ Ax curve, the transverse, 

 linear dimension in the numerator of the 0-diml 

 curvature ratio is taken arbitrarily as the height 

 of the maximum ordinate on this curve. It is 

 measured on the plot, and is expressed in the 

 same units of measurement as the lengths of the 

 1-deg arcs on the circle-of-curvature transpar- 

 encies. Assuming that the height/length ratio of 

 the section-area curve \i 1/4, following the con- 

 vention of Sec. 24.12, plots of 0-diml longitudinal 

 curvature of all such curves are comparable 

 provided the stations on the section-area curve are 

 spaced along its length exactly the same as are 

 the ship stations along the ship line. A 0-diml 

 curvature plot of the A/ Ax curve for the tran- 

 som-stern design of the ABC ship, for the Taylor 

 Standard Series parent form, and for a merchant 

 ship of good design are given in Fig. 67. X. 



49.12 Longitudinal Flowplane Curvature. A 

 longitudinal flowplane around a ship hull is 

 defined in Sec. 4.11 and illustrated in Figs. 4.P, 

 4.Q, and 24.L. This plane is admittedly arbitrary, 

 principally because it is assumed to stand normal 

 to every section line from bow to stern as it 

 crosses that hne. It is likewise somewhat arbitrary 

 to represent a square stream tube along the 

 flowline at the hull as twisting so that one side of 

 the tube, and the same side, lies always in the 

 flowplane. 



As the flowplane is almost never a flat one, in a 

 strict geometric sense, it has to be untwisted and 

 straightened into the flat before its 0-diml longi- 

 tudinal curvature may be measured. This opera- 

 tion involves laying out, on paper, what would be 

 the shape taken by the inner edge of a piece of 

 sheet metal if twisted into the flowplane shape, 

 trimmed to fit the side of a model at the flowline, 

 and then untwisted into the flat. It is a somewhat 

 tedious piece of 3-diml geometry, involving the 

 slightly expanded station lengths as ordinate 

 spacing and the developed lengths of the flowline 

 between stations, measured on the body plan, as 



ordinate increments. Drawing the unrolled and 

 untwisted flowline is followed by a measurement 

 of its 0-diml curvature. 



The process is not described in detail or illus- 

 trated here because it does not take account of 

 the untwisting necessary to get the actual model 

 flowplane into the single plane of the paper on 

 which it is laid down. There can be a large 

 curvature without twist, in the buttocks of a 

 short barge with steeply raked ends, or there can 

 be large twist with small longitudinal curvature, 

 when water flows up and around the bossings on 

 a twin-screw ship. 



Undoubtedly a graphic method could be 

 developed for taking account of both twist and 

 curvature but this should await more definite 

 knowledge as to the hydrodynamic effects of each 

 of these features in creating differential pressures 

 on the underwater hull. 



49.13 Checking and Establishing Fairness of 

 Lines by Mathematical Methods. The measure- 

 ment and plotting of the 0-diml curvature of a 

 selected ship's line, described in Sec. 49.10, serve 

 as a graphic fairness indicator for any hne drawn 

 in the customary way with a ship's curve, spline, 

 or batten. R. Taggart has devised a mathematical 

 method of checking and establishing fairness, 

 entirely independent of any graphic procedure 

 [ASNE, May 1955, pp. 337-357]. Instead of 

 D. W. Taylor's fifth-order polynomial y = a -\- 

 hx -\- ex' -\- dx^ + ex^ + fx^ he found it necessary 

 to employ a sixth-order expression of the following 

 form: 



y = ax -\- hx^ + cx^ -\- dx* + ex^ + fx^ (49.xxiv) 



Furthermore, Taggart uses integrated relation- 

 ships instead of the differential relationships of 

 Taylor to determine the unknown constants. 



The constant first term of the Taylor expression 

 is eliminated if an origin be selected along a 

 continuous part of the ship fine where y and x 

 are simultaneously zero. For the customary 

 waterline this calls for: 



(1) Separate origins at the bow and at the stern 



(2) A maximum value of x where the entrance 

 or run encounters the middlebody 



(3) A slope of zero in the ship line at this maxi- 

 mum value of x 



(4) Offsetting the origin from the plane of sym- 

 metry in the event the half-siding at the bow or 

 stern has a finite value. For convenience the 

 maximum value of x and the half-beam value of 



