The NSMB Mathematical Method
A second method for mathematically developing a set of hull lines for a ship is defined in the paper "Preliminary Design of Ship Lines by Mathematical Methods" by G. Kuiper, published in the Society of Naval Architects and Marine Engineers March 1970 Issue of the "Journal of Ship Reserach" [Hullform Ref 2].
In this paper a ship's hull is broken into three main sections; the entrance (forward), the mid-body, and the run (aft), as sown below. In addition, the paper notes that;
"...for the sake of simplicity some restrictions have been made:
methodology, the fore and aft sections of a ship's hull are described by a series high-order
polynomials, connected by straight lines in the midbody section.
The definition of the fore and aft waterline sections are somewhat similar to the method used in the D.W. Taylor method for the design waterline, where each section is modeled as a high-order polynomial. The primary difference in this method is that the polynomial used here contains a square root term which can give the polynomial a "rounded" end shape.
The primary difference between this NSMB method and the D.W. Taylor & ITU methods are that, while the Taylor and ITU methods use high order polynomials to define a ship's design waterline and then use an alternate set of curves to define a ship's section shapes, the NSMB method uses the same high order polynomial based type of equation to define additional waterlines on the hull. As such, the Taylor and ITU methods can be viewed as defining a hull by means of a mathematically derived design waterline, sectional area curve and section shapes, whereas in the NSMB method the hull is defined by a set of mathematically derived waterline curves.
The equations used to define the fore and aft waterline sections in general take the form of:
y = c0 + c1 * sqrt( 1 - x ) + c2 * x + c3 * x2 + c4 * x3 + c5 * x4 + c6 * x5 + c7 * x6
The coefficients c0 through c7 can be derived for a ship's design waterline by setting certain constraints determined by the desried hull characterisitics. Specifically, since we wish that the above curve close to the ship's centerline at one end but extend out to the maximum beam at waterline at the other end we can define that;
@ x = 1, y = 0
so, 0 = c0 + c2 + c3 + c4 + c5 + c6 + c7
@ x = 0, y = 1
so, 1 = c0 + c1
In addition, since we want the waterline to smoothly taper into "1" at one end (so that it will mate up with, and be "continuously" smooth with the straight line mid section shapes) we can also dictate that at that end the 1st and 2nd derivatives of the curve both equal "0", or;
@ x = 0, y' = 0
@ x = 0, y" = 0
y' = - 0.5 * c1 * ( 1 - x ) -1/2+ c2 + 2 * c3 * x + 3 * c4 * x2 + 4 * c5 * x3 + 5 * c6 * x4 + 6 * c7 * x5
y" = - 0.25 * c1 * ( 1 - x )-3/2 + 2 * c3 + 6 * c4 * x + 12 * c5 * x2 + 20 * c6 * x3 + 30 * c7 * x4
0 = - 0.5 * c1 + c2
0 = - 0.25 * c1 + 2 * c3
If we integrate
the curve to get the area under the curve, since the curve is basically
"non-dimensionalized" then the area under the curve will be equal to
the modified Waterplane Coefficient for that end of the ship, or;
Sum from 0 to 1 = Cwl(for that end) =
c0 * x - 2/3 * c1 * sqrt( 1 - x )3/2 + 1/2 * c2 * x2 + 1/3 * c3 * x3 + 1/4 * c4 * x4 + 1/5 * c5 * x5 + 1/6 * c6 * x6 + 1/7 * c7 * x7
or, Cwl(for that end) = c0 - 2/3 * c1 + 1/2 * c2 + 1/3 * c3 + 1/4 * c4 + 1/5 * c5 + 1/6 * c6 + 1/7 * c7
Another constraint for the design waterline can be set by integrating
the product of the curve and "x" which, since the curve is basically
"non-dimensionalized", will sum to a value of Cwl(for that end) times the Centroid of that end, or;
Sum from 0 to 1 = Cwl(for that end) * CG(for that end) =
1/2 * c0 * x2 + 4/15 * c1 * x * ( 1 - x )3/2 + 1/3 * c2 * x3 + 1/4 * c3 * x4 + 1/5 * c4 * x5 + 1/6 * c5 * x6 + 1/7 * c6 * x7 + 1/8 * c7 * x8
or, Cwl(for that end) * CG(for that end) = 1/2 * c0 + 4/15 * c1 + 1/3 * c2 + 1/4 * c3 + 1/5 * c4 + 1/6 * c5 + 1/7 * c6 + 1/8 * c7
Finally, because of the square root term;
c1 = sqrt( 2r )
And also, by defining a distace e a short distance from x = 1 (far enough away from the end to not be affected by the end radius) we can set the 1st derivative of the curve there equal to the 1/2 entrance (or 1/2 exit) angle of the waterline. Thus;
1/2 entrance/exit angle = - 0.5 * c1 * ( e )-1/2+ c2 + 2 * c3 * (1-e) + 3 * c4 * (1-e )2 + 4 * c5 * (1-e)3 + 5 * c6 * (1-e)4 + 6 * c7 * (1-e)5
Putting all this information together we end up with eight "equations" and eight "unknowns" which we can use the Matrix Inversion andMultiplication tools in a spreadsheet like Excel to solve.
Similarly, if we want to develop a similar curve for another waterline either above or below the design waterline, we can do so by substituting in the values for;
Because the actual overall location of the ends of the various waterlines will vary due to the bow and stern profile shape, as well as the extent of the flat of sides at a given waterline, and the max beam may likely change as well, it may become necessary to "modify" a given Cwl, CG, ie and/or other term to adjust it to make the inputs and constraints consistant witht the internal math of the process, which assumes each waterline extends from 0 to 1 with the beam also extending from 0 to 1, and where the Cwl(for that end) and CG(for that end) are based on the min and max Lengths of that specific waterline. to this end the mehtodology provides a means to scale the inputs and outputs for specifc waterlines as dictated by the local lengths and beams, etc.