I started off wondering why my boat develops weather helm as she heels. Must be the increasing asymmetry in the heeled waterplane, I said to myself. So let's find a way of seeing this waterplane... If the hull sections are approximated by circular arcs (which would be quite agreeable for designs such as the TS2 and Ikon), then a simple spreadsheet could do the calculations and graph the results.
It turns out, of course, that the resulting spreadsheet ("design.xls", 97kb) is effectively a hull design program that is restricted to arcs of circles for the sections. (Now updated to provide for any LWL, not just 1000mm as required for an IOM. Design your next Marblehead this way.) Not too much of a restriction, perhaps, given the success of the TS2 and similar boats. It is a hull design program because, to get the heeled waterline, you first need to make decisions about the sort of rocker profile you want, the way the sections flatten towards the transom, where you want maximum beam, and so on.
The basic idea is to take a set of circular arcs, and distribute them over the length of the boat to yield a set of hull lines. This in turn raises two questions: where to place the circle centres, and what radius to give each circle. Part of the answer to the circle's radius is provided by the rocker profile -- the sort of static draft you want the hull to have. Part of the answer of where to place the circle centres is provided by the sort of beam you want at mid-section and the sort of flatness of the underbody you want at the transom.
The spreadsheet provides several ways of generating a rocker profile, where each method can be varied by using parameters. The rocker profile graph looks something like the following. The particular profile shown is a "reverse parabolic". Other profiles include circular and hyperbolic. Of course, if you don't like these, you can specify your own.
Then, the spreadsheet provides a number of ways to distribute the circular arcs along the length of the hull. The result is a set of circles whose lower arcs are the hull sections. The spreadsheet calculates the circle centres and radii; you have to plot them onto paper to see the what you have. The spreadsheet methods to distribute the circle centres are parabolic, hyperbolic, circular, and linear. Again, if you have your own ideas, no problem...
The key calculations are to compute R, the radius, given the rocker draft d and the centre of the circle, as well as L, the semi-beam. From these, the area of the immersed section is calculated.
The immediate result is an approximate plot of the hull sections. This is approximate because the graphing functions in Excel do not lend themselves to CAD-type work, but the plot gives a reasonable impression of the resulting design.
To help with finding a set of good lines, a plot of immersed section areas is provided. This helps visualise how the underwater volume increases and hence the sort of drag which results from pushing the water to one side or the other.
Then, you can specify an angle of heel, and the spreadsheet calculates and plots the waterplane. This example is for a nearly-symmetric hull, characteristic of a narrow beam, narrow transom design. Notice that the heeled waterplane shows a leeward bulge (that provides some of the righting moment), and that it is not symmetrical.
Here is an example of a heeled waterplane that is characteristic of a wide transom "skiff" type of design. For the transom to be wide, the design must obviously also have a wide beam. What is significant, however, is that having a wide beam is not what is important; the "skiff" type IOM design is interesting because of its wide transom. As can be seen, almost all of the heeled waterplane is offset from the yacht's centre-line, and this contributes towards an increased righting moment. Notice that the heeled waterplane is almost perfectly symmetric. Finally, notice that the heeled waterplane is "slewed"; its axis of symmetry is at an angle to the course being sailed. I used to think that the slew contributed to weather helm, but it turns out I was wrong. Gary Cameron has very kindly straightened me out on this, and the details are provided on the "Balance" page.
The spreadsheet calculates a number of other results as well -- volume displacement, longitudinal centre of buoyancy, wetted surface area, change of trim when heeled, angle of slew, increase in righting moment when heeled, and so on.
©2011 Lester Gilbert