Heeling Arm Curves
In the "Flat Plate" stability model, wind heeling force diminishes with heel. Thus the force rotating the boat clockwise in the illustrations is diminishing at the same time that the Righting Moment the boat is developing to resist it by creating force in a counter clockwise direction is increasing. Here is how we look at that balance graphically.
If we take the upright, i.e. at zero heel angle, Heeling Moment, (Wind pressure per square foot x windage area x heeling arm) and divide it by the vessel's displacement, we get a number called a "Wind Heeling Arm". This is an imaginary distance that can be directly compared with the GZ distance graphed on the Righting Arm Curve. There will be a different Wind Heeling Arm for each wind velocity. Assuming for the moment that heeling moment doesn't decrease with heel, the graph looks like this:
If there were not reduction in wind heeling force as the vessel heels, this graph would show a vessel with a sail plan that would heel it 6.6 degrees in a 10 knot breeze, 14.5 in a 15 knot wind, and 26.7 in 20 knots. 21.3 knots would heel her right to the peak of the righting arm curve at 40 degrees. This means that any stronger wind would capsize her. Note that wind force increases as the square of the velocity so that a 20 knot wind exerts four times as much force as a 10 knot wind.
However, according to the flat plate model, and in real life, heeling force decreases as the sail plan assumes an angle to the wind due to the vessel heeling. Let's look at just one of the heeling cases above, that for 20 knots of wind. The actual graph would look like this:
There would be a similarly shaped curve for each of the other wind velocities. The theoretial predicted heel angle in a steady wind is the angle at the point where the two curves cross. Each curve starts at the value shown in the first figure and is constructed by multiplying its value at each heel angle by the Cosine squared of the angle. Since a flat plate has no wind resistance laying down and the Cosine of 90 degrees is 0, there is no wind heeling force at 90 degrees. "Wait a minute", you may be saying, "about half of the hull is sticking up when a boat is on it's side and there is wind pressure on that so there should be some wind heeling value at 90 degrees." Well, that's just another thing you know that the authors of textbooks and the Coast Guard graduate naval architects who write stability regulations don't. Pat yourself on the back. By the time heel angle reaches 90 degrees however, the theory is getting lost in the excitement. Everyone is screaming and floundering around trying to get the boat back on its feet or, in the case of the craft show above, frantically trying to disengage themselves from it before it sinks because it isn't going to come back upright again.
Wind tunnel tests on models of vessels conducted at the Wolfson Unit in England have actually shown the Cosine squared curve to be pretty accurate for fore and aft rigged vessels. The curve bulges up a bit more for square riggers due to the different geometry of their sails but it is still a workable approximation.
Next: Why the Stability of Large Vessels can be Treacherous