In 2008 I wrote a short paper on a mathematical method of developable hull design (which is included in this blog, see entry of Dec. 2008). I tried to keep it short and discuss only general principles, but some subjects are difficult to understand without providing examples. Without any examples, I dont think it was very meaningful for most readers. Recently the subject came up again, concerning the difficulty of developable design and the paucity of explanatory material. Last night, I wrote a short paper concerning a stepwise approach and actual sample calculations involved in creating a hull shape. The numbers used are from the runabout building in my shop currently. The paper is not a comprehensive "how to" but hopefully will make the subject more easily understandable.
The entire shape is not yet designed, but sequence and type of calculations needed should be understandable. The results, when finished, are full-size measurements in three dimensions with fair curves and excellent accuracy using simple tools. Enough offsets are generated that all you have to do is connect the conveniently-spaced dots.
Sample computations for Developable Design
To illustrate the generation of many exact points along the chine curve, the following is a list of some coordinates of the chine at 7” intervals for the anterior chine of the hull I am now building: (0, 0, 24.48), (7, 3.2, 22.56), (14, 6.3, 20.7), (21 ,9.2, 18.96), (28, 11.9, 17.34), (35, 14.4, 15.84), (42, 16.7, 14.46) ,.…. until (119, 28.8, 7.2). This series is generated from the parabolic curve Y = 28.8- (119- X)squared / 490 and also Z = 0.6(28.8-Y) + 7.2 and is valid for values of X between 7 and 119.
Where did these equations come from? I want a boat about 6’ by 18’, so I picked a length of about half that which I could evenly divide into 16 segments [more segments = more accuracy]; 16x7 or 112” will suffice, next add a short straight 7” segment onto that. Have you ever noticed that when you bend a batten, the curve does not extend all the way to the end of the board? At the batten end there is no fulcrum to apply torque. Thus, last few inches have no bend unless confined along its entire length. So I always add a straight segment at the end of every curve. A 6’ beam gives a 36” half beam; minus some width for flare of the topsides (4”) and an allowance for a chine flat (3.2” at maximum point) and you end up with 28.8” chine beam which is 25.6” of camber in the 112” length and 3.2” of offset along the 7” straight end segment.
To create the forward keel projection, I most frequently use a parallel projection. A conic projection, with the apex of the cone forward, will create sharp curvatures in that area. It may be what you want, but it will also be hard to plank. A conic apex amidships will give a mild curve at the keel, if that is what is desired. The parallel projection has a controllable curvature and is easy to calculate. We already have defined a 28.8” chine beam, and picked a 7.2” deadrise to go with it [based on estimated displacement]. So deciding on the slope for this parallel projection from chine to keel only involves selecting an X intercept from the maximum chine point (119, 28.8, 7.2). The further forward we select this point, the more pronounced will be the forefoot of the keel with sharper curvature. I chose a point 52.5” forward of maximum chine beam. 52.5/7.5 = 7; 28.8/7.5 = 3.84; 7.2/7.5 = 0.96; thus we have our slope, X: Y: Z=7: 3.84: 0.96 which coincides with the 7” segment intervals.
Now all anterior keel intercepts can be calculated to define the keel. At the keel, Y=0, and we solve for X and Z. X@keel= X@chine- (7/3.84)Y@chine and Z@keel= Z@chine– (0.96/3.84)Y@chine. Aft of the point X=119”, the keel is straight and Y=0 & Z=0 for the keel. After finding the shape of the keel, we next move on to the shape of the transverse frames below the chine. To calculate transverse frame shapes below the chine, the same slope or ratio is used. We set X = 14, 28, 42, etc., or whatever other frame locations desired, and solve for Y and Z. For every 7” forward we project a chine coordinate, the Y dimension will decrease by 3.84” and the Z dimension will decrease by 0.96”. Simple math creates the entire shape of each frame. Just connect the dots.
Creating topsides with some flare to the bow, transitioning into a tumblehome stern, seems to be best accomplished with a conic projection forward, linked to another conic apex further aft to create the transition, then a parallel projection extending aft to finish the tumblehome contour. Conic projections involve finding a third point on a line give two defined points. Rather than discuss an entire design, I will list selected apices and show sample calculations.
The apex of the first cone was selected at X, Y, Z = (56, 39, 70.2). This will give a bow angle which matches the forefoot of the keel and provide moderate flare to the topsides not to exceed a 6’ total beam. A sample calculation would be to calculate the Y and Z intercepts at the frame location X = 98 for a line between the apex and the chine coordinates (119, 31.9, 7.2). The calculation is Y = 39– (98- 56) (39– 31.9)/ (119– 56) = 34.27 and Z = 70.2– (98– 56) (70.2– 7.2)/ (119- 56)= 28.2. The relation is that the change in any one coordinate of a point on a line is proportional to the change in any other coordinate. Since we choose our X intervals, we can then find Y and Z.
When calculations are complete for X between 0 and 126, we select a second apex (91, 35.5, 38.7) which lies on the ruling line, halfway between the first apex and the point of maximum chine beam (to the outside of the chine flat) which is (126, 32, 7.2). This new apex is then used to calculate points aft to the chine location (189, 32, 7.2). From there a parallel projection is used with the slope 7: 0.25: 2.25 and defined points every 7” along an extended chine equation to X= 294. Although the actual chine ends at X= 213.5, the extended portion will determine the shape of the tumblehome at the stern when projected forward. As more curvature is included in this extended curve, the tumblehome will also increase.
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