Sec. 49.9 



MATHEMATICAL LINES FOR SHIPS 



195 



where x was the distance of any point on the 

 forebody surface reckoned forward of the forward 

 end of the middlebody, and y was the radius from 

 the airship axis to that point. When x equaled 

 zero, at the junction point of the entrance and 

 the middlebody, the second derivative d^y/dx^ 

 for the outline equaled zero, which meant that 

 the curvature was also zero. Since the straight 

 contours in the middlebody had zero curvature, 

 those of the entrance joined them in what might 

 be called a continuous transition. A 0-diml curva- 

 ture plot of the outline of a longitudinal axial 

 section would show no break at that point. 



In Reports and Memoranda 256 of the (British) 

 Aeronautical Research Committee, dated June 

 1916, W. L. Cowley, L. F. G. Simmons, and 

 J. D. Coales report as follows: 



P. 167. "Previous reports and preliminary experiments 

 for the present report all showed that the air resistance of 

 a strut was extremely sensitive to slight changes in the 

 form and radius of curvature, especially in the neighbor- 

 hood of the maximum ordinate. For this reason each strut, 

 of a series in which one proportion only is to be varied, 

 must be made with fair accuracy in those dimensions 

 which are to be kept constant." 



P. 170. "The results of the investigation appear to show 

 that the air resistance of a strut depends very greatly upon 

 the shape of the fairing in the region in front of the maxi- 

 mum ordinate. In a round-nosed strut the change of curva- 

 ture and slope in passing from the nose to the fairing piece 

 should not be too rapid, a condition which causes the 

 maximum width to be some distance behind the center of 

 curvature of the nose." 



49.9 Notes on Longitudinal Curvature Analy- 

 sis. For determining and analyzing the curva- 

 ture of any fore-and-aft ship line, such as the 

 designed waterlines discussed in Sees. 4.4, 4.5, 

 4.7, and 24.13, there are several methods besides 

 the semi-graphic one described in those references. 



Assuming an origin of coordinates at any con- 

 venient point along the ship centerplane, with 

 abscissas x parallel to the x-axis and ordinates y 

 measured transversely in the plane of the selected 

 line, the slope of that line with reference to the 

 X-axis is dy/dx. This may be measured as a 

 natural tangent or as a function of an angle, 

 where dy/dx = tan 6. 



The rate of change of slope, called by D. W. 

 Taylor the "acceleration" a(alpha) in his 1915 

 mathematical-lines paper referenced in Sec. 49.3, is 



The radius of curvature Kc of any curved line, 

 from standard reference works on this subject, 

 is expressed as 



dx' 



[I + tan' 



MSI 



d^ 



dx' 



(49.xxi) 



(49. xx) 



= 5- sec B 

 tic 



The absolute curvature is l/Rc ', this is dimen- 

 sional because Re is dimensional. Strictly speak- 

 ing, an irregular curved line has no definite radius 

 of curvature. However, at any selected point it 

 has an effective Re equal to that of a circle, called 

 a circle of curvature, which coincides very nearly 

 with the given curved line at the given point. It 

 is possible to plot the dimensional curvature of 

 ship waterlines on a basis of length along the 

 fore-and-aft axis by using the reciprocal values 

 l//?(7. A. Emerson has done this by employing the 

 first and second differences of the waterline offsets 

 at 40-station intervals, each equal to 0.025L 

 [INA, 1937, Fig. 6, p. 178; unpubl. Itr. of 11 Jan 

 1952 to HES]. Plotting 0-diml longitudinal 

 curvature is described in Sec. 49.10. 



Any of the foregoing methods involving dy/dx 

 or ^y/dx is satisfactory only if offsets, slopes, 

 and rates of change of slope are taken for at 

 least 40 equally spaced stations along the length 

 of a ship. No method is satisfactory unless the 

 waterline itself is carefully and accurately drawn. 



From Eq. (49.xxi) preceding it is apparent by 

 mspection that Re approaches infinity as the 

 second derivative d'y/dx^ approaches zero. This 

 is equivalent to saying that the absolute curvature 

 l/Rc approaches zero with the second derivative. 

 Since the curvature of a straight line is zero, a 

 suitable transition from a curved ship line to a 

 straight one, such as the parallel portion of a 

 waterline, is marked by a diminution of d^y/dx' 

 toward zero at the junction ^vith the straight line. 



Proper transition is an acute problem in the 

 laying of tracks for railway cars and trains. If a 

 straight section of track, called in railway parlance 

 a tangent, were joined directly to a curved section 

 of track having a constant radius, the transverse 

 acceleration at the junction would be so great on 

 a high-speed train passing from the straight to 

 the curved section that the train would leave the 

 rails or the track would be torn up. Railway 

 surveyors have developed several acceptable 

 methods for making this transition between 

 straight and curved portions of track. One of them 

 involves the use of a second- or third-order parab- 



