Stiffness (AMYA MY #173)

Stiffness

by Lester Gilbert and Graham Bantock

From time to time, Graham Bantock publishes a comparison of various IOM designs. One of the interesting aspects of a design is its righting moment, expressed as its “arm” or “gz” at 40 degrees of heel. With the latest publication of his data, I thought now would be a good time to understand righting moment better.

Figure 1 Geometry of a heeling yacht

Balance

Consider a heeled boat in balance or equilibrium, the force of the wind on the sails being balanced by the lump of lead at the end of the fin, as illustrated in Figure 1. The righting moment produced by the weight of the boat depends upon the amount to which the boat’s transverse centre of buoyancy moves to leeward as the boat heels. The point on the boat’s centreline directly above the transverse centre of buoyancy is known as the Metacentre, M for short. The Metacentric Height, GM, is the distance of this point above the centre of gravity, G. For normal IOM hull shapes and normal heel angles the metacentric height varies very little and it is reasonable to consider M as a fixed point. The righting moment, then, is the product of the boat’s displacement and the distance GZ, the righting “arm”, which depends upon the sine of the heel angle. It is obvious that the righting moment increases as the heel angle increases.

What might be less obvious is that the effect of the wind on the sails causes heel, but the effect of increasing wind strength progressively weakens at extreme heel angles. You can do a thought experiment and imagine the boat almost completely blown over; at this point the wind could double in strength but the boat would hardly heel over any more. Why would that be? There are two reasons. One is that the wind speed varies with height above the water surface.

Generally speaking, the relationship between wind speed and height above the water surface is logarithmic. (See the “wind gradient” topic at http://www.onemetre.net/Design/Gradient/Gradient.htm.) At zero height above the water, then, wind speed will also be zero. This is simply another example of the boundary layer effect that allows dust to remain on your car even though you have driven it fast. Two is that the sail area projected to the wind becomes vanishingly small as the boat is blown over, proportional to the cosine of the heel angle.

Arm

The ability of the boat to resist heeling depends on the depth of the vertical centre of gravity (VCG, highly dependent on the length of fin), and also upon the metacentric height (which depends on the beam of the boat). In the IOM class, since almost all boats have a draught of 420 mm, almost all bulbs are around 2.35-2.4 kg in weight, and almost all boat and rig construction weights are the same, the VCG is virtually fixed, and is approximately 90 mm (No 1 rig) or 225 mm (No 2 rig) below the waterline. It is therefore the variation of the metacentric height that provides the differences in the righting moments of boats and this depends upon the differences in their beams. As a thought experiment, we can imagine an IOM hull that is a simple cylinder. Such a hull would have a metacentric height of 0, since its metacentre would coincide with its centre of buoyancy, but it would have a righting arm, approximately 56 mm.

M is about 20 mm above the waterline for narrow hulls, about 40 mm for my Pikanto, about 70 mm for the Lintle, and about 90 mm for the TS2. For the canoe body (ie ignoring buoyancy of fin, bulb, or rudder) the depth of the vertical centre of buoyancy (VCB) is about 0.35 T, and the distance BM (VCB to M) is about WLB²/24 x T x VCB, where WLB is the waterline beam and T the canoe body depth.

We can compare the righting moment of two boats by calculating their effective righting arm at some appropriate angle of heel, often taken at 40 degrees. These are listed in Table 1, based upon Graham’s data. The widest design is the TS2, and it has the largest arm, while the narrowest design is the “Scharmer” with the lowest arm.

Table 1 IOM design arm

The arm has a close (but not perfect!) relationship to the hull beam. Variations are largely due to the degree of flare/tumblehome, and the fineness/fullness of the bow/stern. The following graph, Figure 2, plots the arm against the hull’s waterline beam, again based upon Graham’s data.

Figure 2 Righting arm versus waterline beam for a variety of IOM designs

Heel

My question at this point is, “So what?” What practical difference is there between the 158 mm arm of a TS2 and the 134 mm arm of my Pikanto, or the 117 mm arm of the Scharmer? One way to look at this is to ask how much these different designs would heel at a given wind strength. The “average” design here is the Pikanto, and I calculated the notional wind strength that would cause a Pikanto to heel at approximately 40 degrees. I then calculated how much the other designs would heel in this notional wind. In order to do this, I estimated the centre of effort of the sail plan as approximately 740 mm above the waterplane.

The results are shown in Table 2 and illustrated in Figure 3. The diagram makes it visually clear how much stiffer the TS2 is by comparison with the other designs.

Table 2 IOM design heel

Ignoring the extreme designs (TS2, Scharmer), what practical difference is there between the Ikon’s heel angle of 37.9°, the Pikanto’s of 40.1°, and the Tonic’s 42.6°? You would be hard pressed to see these differences while sailing, illustrated in Figure 3, even on flat water.

Figure 3 Heel angles visualised at constant wind

Effective and equivalent bulb weights

A second, and perhaps more impactful way to look at this is to ask how much a lower arm and a higher heel angle means the boat has a lower effective bulb weight. That is, if we imagine a TS2 with a nominal 2.4 kg bulb that heels at 35.4° in a given wind, what is the “equivalent”, lighter, bulb that would see it heel at 43.8° like the Scharmer design? It turns out that this would be a 2.17 kg bulb. Another way of saying this is that a TS2 would heel at 43.8° in the same given wind if it had a 2.17 kg bulb. My Pikanto, an “average” design, heels at 40.1° in this wind. It has an “equivalent” bulb of 2.27 kg by comparison with a TS2 – that is, a TS2 would heel at the same 40.1° in the same wind if it carried a 2.27 kg bulb. Ah! The results for other designs are shown in Table 3.

Alternatively, let’s ask how much heavier a bulb would need to be in order to make a boat as stiff as a TS2 with a 2.4 kg bulb. The results are shown in Table 3 and illustrated in Figure 4. The required bulb weights run from the 2.4 kg for the TS2 to the 2.70 kg required by the Scharmer. If I wanted my Pikanto to match the stiffness of a TS2, it would need a 2.55 kg keel, 150 g heavier than the nominal 2.4 kg of the TS2. Now that is a practical difference I think I understand! Interestingly, these differences are more modest than I thought they would be.

Table 3 IOM designs’ “equivalent” bulb weights

Figure 4 Required bulb weight to match TS2 stiffness