How to locate the fin and mast of a model yacht
This is a problem that is not finding a solution in the model world. Too many boats are set up by guesswork. I think that it is possible to do much better than guesswork for models with aerodynamic keels. I do not know what to do for models without aerodynamic keels. (See section 5 of this text.)
Every sailing rig can be thought of as producing a single net force that has two components, one that drives the boat forwards and the other that acts across the boat and makes it heel to leeward. We need the one that drives the boat to overcome the resistance to motion of the hull and of the various parts of the rigging. We do not want (but cannot avoid) the one that acts to leeward because it has to be resisted in some way.
I have drawn figure 1 for a racing yacht with the ubiquitous Bermuda rig and an aerodynamic keel that I shall call a fin. I have shown the yacht with the rig fully sheeted in and beating. In the end view I have shown the net forces acting on the rig and on the fin. They are equal and, vertically, they are a long way apart. These two forces act together to make the yacht heel to leeward. To counteract this the weight fitted to the end of the fin swings out to windward.
In the side view I have shown where the net force on the rig acts on the rig and on the hull through the mast and rigging. I have also shown where the force on the fin acts on the fin. I have drawn verticals through these two points and shown them with the vertical for the fin ahead of the vertical for the rig by a small distance . For all models the distance is less than an inch.
Now we can see the problem for model makers. In a nutshell we need to know how to find these two points of action on the rig and the fin so that we space them properly and know where to put them on the hull.
This means that we require some sound basis for our choice. It must be obvious that whereas I have drawn my diagram for a yacht beating into the wind there are all the other points of sailing when the sails will be in quite different positions and the net force on the sailing rig be in all sorts of directions. It looks to be impossible. Fortunately for us it is essential that the boat must tack through head to wind as easily as possible and this takes precedence over every other point of sailing. This is why I drew my yacht as if it was beating. If it is to turn easily to windward the force on the fin must be ahead of the force on the sailing rig by some small amount so we must design for this condition.
This means that first we have to put a figure to the distance . Clearly if we give a large value the boat will be forever wanting to turn to windward and require rudder correction all the time. That will produce unnecessary drag from both the rudder and the fin. What we want is the boat to turn gently to windward when it is beating up to the wind and the rudder is centred. Then the boat has weather helm.
We have to use whatever experience we can bring to bear to choose a value for . My best guess is that should not be greater than an inch for big boats and perhaps a half inch for small ones.
This information is only of value if we can locate the two points of action with some confidence. This is possible in both cases but it is best if I deal with the fin and the rig separately and I will start with the fin. I shall call these points of action centres of pressure to fit the usual nomenclature.
1 The centre of pressure of the fin
I think that there is a good case for fitting an aerodynamic fin with streamlined ballast on the end of it to all model sailing boats. (and to some power boats) It certainly simplifies the problem of setting up a boat.
In the introduction I ignored the hull altogether but now I need to make the case for using a fin and lead to cope with the transverse force and the stability.
For what may be several thousand years the transverse, leeward force was resisted by the hull moving with its centreline at a fairly large angle to the windward of the course of the boat. The boat appeared to be sailing both forwards and sideways and it was said to make headway and leeway. This meant that the flow under the hull was skewed to the hull and many hulls had bits added underwater to increase the resistance to this skewed flow and so reduce the leeway. It is evident that the hull was not designed to operate with leeway but to be altered in retrospect to reduce the leeway to an acceptable angle.
So what is the problem for the model maker? For the builders of scale model boats the laws of physics throw up two major problems. The first is that whilst the model boat must work in the same wind speeds as the full size it moves through the water very slowly at less than two miles an hour whereas the full size would have made 6 or 7 mph. This means that our model hull cannot generate a large enough transverse force to resist the transverse force produced by the sails. The second is that the model will not have enough stability to stay upright with all its rig in normal winds. It is usually necessary to carry external ballast under the hull to give sufficient stability to limit the heel and for the model to look realistic when sailing. This ballast needs to be well below the scale keel and has to be attached in some way. We have two problems to solve.
The evidence from racing yachts is that the best thing to do to solve these problems is to attach the external ballast to an aerodynamic keel fitted to the hull, that is, to a fin. This fin is so much more effective than a hull that it can resist the transverse force with a much smaller leeway than a hull and it replaces the hull as the principal means of resisting the transverse force. At the same time it gives an extremely handy place to carry the underwater ballast in the form of a streamlined weight. (Some modellers reject the addition of a fin and lead to a scale model out of hand. They regard it as an admission of failure when in fact it is an acceptance of the laws of physics. I do not know of an alternative solution except to fit floats to the top of the rig and have a rescue line to hand.)
So what follows is for models fitted with a well-designed aerodynamic fin and I want to find out where the force is exerted on such a fin.
We can look to aeroplanes. If it were to be fitted to an aeroplane the fin would be one half of a pair of symmetrical sectioned wings. Aeromodellers who have flown models fitted with such wings know that the aeroplane must balance fore and aft if it is to fly properly and you see them supporting the aeroplane on two finger tips under the wings to check the balance. They will be looking for the balance point on wings of constant chord about 35% to 40% of the chord back from the leading edge and most tend towards 40% because it makes for a very responsive model. What has been done by this balancing is to get the single force produced by the wing in level flight aligned with the weight of the model acting through its centre of gravity. In other words the centre of pressure of a constant chord wing is 40% back from the leading edge. We can use this information to find where the force acts on the fin of a model yacht.
Going back to the fin, the force acting on it would be at the 40% position if it were to be rectangular in shape. Most fins are not rectangular but taper from being wide at the top. If we chose to make a fin to the profile in figure 2 we could be confident that the force on the fin would act at a point, that is usually called the centre of pressure, that is somewhere on the vertical line shown that joins the 40% points at top and bottom. The resulting profile is quite acceptable to the eye.
So, because the fin is going to provide very nearly all of the transverse force, if we attached such a fin to a hull, we would know where the force on the fin acts on the hull. Furthermore if there is any transverse force at all generated by the hull it will act at about 40% of the length of the waterline from the forward end. So, if the fin is attached at this 40% point we know with some confidence where the transverse force on the hull acts and therefore on the yacht as a whole. It is then quite straightforward to find the vertical through the point of action of the force on the fin. I shall call that point the centre of pressure and denote it CoPF.
2 The centre of pressure for the rig
Now if we are to get the force on the sailing rig in the best position relative to the fin we must find some way of knowing where the force acts on the rig, that is, the centre of pressure of the rig CoPR.
We need a simple strategy. We know that we need to consider the sailing rig sheeted in for beating and others have seen that the centre of pressure of the rig will not be far away from the centre of area of the rig when sheeted in. We can find the centre of area of the rig quite easily and we can narrow down the distance between the centre of pressure and the centre of area.
Whatever sailing boat we may be interested in, it is normal to draw a side elevation of a sailing boat showing the rig with its sails laid out flat in the correct position on the rigging. It is called the sail-plan. At one time the-sail plan had two other sets of information. It was usual to show the area of each sail and the centre of area of each sail. This information is all that is needed to find the position of the centre of area of the whole rig as it is laid out in the sail-plan. Anyone can do it.
We have seen that aeromodellers find that the centre of pressure on their wings is at 40% back from the leading edge for straight wing of constant chord. As the centre of area of that wing is at 50% of the chord from the leading edge one might also say that it is 10% in front of the centre of area. The fact is that the centre of pressure on wings of constant chord is always 10% to 15% and in front of the centre of area. If this can be transferred to sails we could get a better idea of the position of the centre of pressure of the rig from the position of the centre of area of the rig. This appears to be valid and so we need to know how to find the centre of area of the rig.
Figure 3 shows the sail plan of a sailing barge as it was in 1936. At the top right the areas of individual sails are given and the centres of area are on the sail plan as black crosses.
Now we do not want details of individual sails but the centre of area of the whole set of sails as they are laid out here. In order to find that we need to do a little drawing. It is shown in the figure 4 of Pearl. The added construction is in red. First we must set up a vertical reference line. It could be anywhere but I have drawn it through the junction of the top mast stay and the bowsprit. Now we must multiply the area of each sail by the distance of its centre of area from the reference line as shown in red and add all the resulting figures together. Then if the sum is divided by the total area of the sails we shall have the distance of the centre of area of the whole rig, as laid out in the sail plan, from the reference line.
Of course the centres of area may not be given and it may be necessary to find the areas and the centres of area of the separate sails. Do not try to do this for the shapes of the sails as they stand. Divide the sails into triangles and use the fact that regardless of shape the area is equal to where and are as shown in figure 5 and the centre of area is at the point of intersection of lines joining the apexes to the centres of the opposite sides as shown in figure 6. Then for Pearl instead of having eight small areas to deal with there would be 10.
In my experience many modellers have little confidence in their ability to do the arithmetic needed to find the position of the centre of area of the sail plan even when they are given centres of area of the sails and their areas. They feel more confident when using a practical method. The method used has its roots in the school physics exercise to find the centre of gravity and, by extension, the centre of area of a piece of card having some arbitrary shape. The method involves mounting the card on a pin so that it swings freely and using a plumb line from the pin to draw a vertical line on the card. If this is done two or three times the lines cross at the centre of gravity of the card which can be taken to be the centre of area. We can use this method but our card would have the shape of the sail plan and we only want to know the horizontal position of the centre of area.
I have never attempted to use this method except at school and it seemed to me that I should try it to see what the practical problems might be and how the result compared with the calculated position. I have some card 0.05" thick or 1.18 mm thick. I took the sail -plan of Pearl in figure 3 and enlarged it to fill an A4 sheet and printed it twice. I stuck one copy to the card and cut the sails out using a band saw. Knowing that I would have to re-assemble the sails on the second copy I used overlaps to minimise the cutting. The cut sails are shown in photo 7. There was no need to separate the main and the mizzen from their top sails nor to separate the three jibs that are joined at the overlaps.
Somehow these "sails" have to be re-assembled and I did that on the second copy.
Photo 8 shows the card "sails" back in their correct positions with the backing paper cut away to leave the rig tied together but with a tab at the top for the pin. The sail plan is resting on a piece of stone coloured card. For the record the paper was 80 gram copying paper and one whole sheet weighed 3.1 grams and the card "sails" weighed 13.2 grams. It looks to be very flimsy but it worked without difficulty.
Photo 9 shows the sail plan suspended on a pin with a black head that was pushed into the card that was held upright in two toolmakers cramps. The sail-plan swung very freely and after a few attempts I found a position for the pin where the sail-plan settled down with the luff of the main vertical as indicated by the try square. It was very easy to do. I could locate the vertical through the centre of area of the sail-plan.
The resultant position is shown by the short black line on the suspension tab. I did this before I attempted to calculate this position. When I did make the calculation the two agreed exactly which was a better result than I expected. So this small model that is only 230 mm (9") long is perfectly adequate.
This does not seem to be very onerous and anyone can do it. We do not need to use the full size sail plan but the sails must be drawn to scale to a size that is convenient.
There is another piece of information that can be extracted from the sail plan of Pearl. Sailing barges were all fitted with leeboards to act in exactly the same way as the fins on racing yachts. They evolved to have very respectable cross sections that were very like a modern thin symmetrical aerofoil. The leeboards were not lowered to 90º so that there was always tension in the lowering chain. I have added a leeboard in the lowered position to part of the sail plan of Pearl˝ as shown in figure 10 and I have included a vertical line that passes through the centre of area of the sail-plan. I have also shown the likely position of the net force on the submerged part of the leeboard. That point is just ahead of the centre of area of the rig. This is exactly what we might expect. That point is at 41% back from the stem and no doubt the raising or lowering of the leeboard and moving the force on the leeboard could have trimmed the running of the barge. This was all found by trial! Had this been of a 1/24 scale model with a hull length of 42" the horizontal distance between the points would be about 1" and the distance between the mast and the centre of area would be 4.5".
Sorting out the distances
For any sailing boat if we are to locate the mast and the fin on the hull we shall have to go through this same procedure of finding the centre of area of the sail plan relative to the main mast and finding the horizontal distance between the centre of area of the rig and the centre of pressure of the rig.
I have drawn a racing yacht in figure 11 and on it I have marked the centres of area of the two sails and used these to find the centre of area of the rig as a whole. They are marked as CoAM for centre of area of the mainsail, CoAJ for the jib and CoAR for the rig.
However what we want is the centre of pressure for the rig and we cannot do that directly. What we can do is use the wing information and place the centre of pressure ahead of the CoA of each sail by, say, 10% of the width of the sail at the centre of area. This I have done but I still need to locate the centre of pressure of the rig. Then the crucial piece of information is in the following statement. It turns out that the CoPR is ahead of CoAR by less than the distance CoPM to CoAM and more than CoPJ to CoAJ. This means that we can narrow down the possible error in the position of CoPR.
Figure 11 is too small to let us see the positions of the various lines easily so I have drawn the important part of the diagram to a much larger scale as figure 12. The point that we know for certain is the centre of area of the sail-plan, CoAR. I have argued for the distance from CoAR to CoPR to be a bit less than 10% of the width of the mainsail at CoAM but it could be more but not more than 15% so we could just leave this length as 10%. We want this yacht to have weather helm and, for that, the CoPF must be in front of CoPR. I have no way to put an authoritative figure to this but my guess is 1" for a big boat and 0.5" for a small one.
With these figures, and the fact that the fin should be at 40% of the waterline, all the parts of the yacht can be located.
We need to have some idea of the magnitudes of these distances. The width of the mainsails at the centre of area are, for my A boat, my Marblehead and my 27" boat are about 17", 11" and 9" so using the 10% figure the maximum distances of the centre of pressure ahead of the centre of area are 1.7", 1.1" and 0.9". For my sprit-sail barge the comparable width of the mainsail it is about 18". Of course for a given sail-plan the distance between the centre of area of the rig and the centre of the mast can be measured separately.
If now the allowance is made for the weather helm the distances between CoAR and CoPF become about 2.7" for the A boat, about 1.85" for the Marblehead and about 1.4" for the 27" yacht.
Experience suggests that these figures are not critical but I would err on the side of a small increase rather than a decrease. It is very important to go able to go through head to wind.
3 Other problems
One needs only to look at figure 9 to see that there is a practical problem in design because the mounting for the fin and the mast tube are close together. This becomes particularly acute for Marbleheads. These yachts have evolved to use rigs of very high aspect ratio and the main sails are very narrow so that the centre of area of the rig is very close to the mast. The fins have also become very long and narrow and, of course, very thin. As a result the only satisfactory way to mount the fin is to let it have a wide tongue that extends into a box inside the hull. The fin box and the mast tube need to occupy the same space with the mast tube ahead of the fin. This becomes a problem to tax the ingenuity of the designer.
4 Scale and semi-scale rigs
I have analysed this problem for the racing yacht but the same problem arises for scale models and semi-scale models of all sorts. Semi-scale yachts with Bermuda rigs can be dealt with like racing yachts. For a scale model where a rig has two masts it is sufficient to find the distance between the centre of area of the rig and its centre of pressure by using 10% of the width of the mainsail at the level of its centre of area. But here we cannot locate the fin by using 40% of the waterline because the mast position is fixed by the prototype design, it must be located relative to the centre of area of the rig.
5. The problem of scale hulls
I said that I do not know what to do with models without aerodynamic fins. I had something to say about this in the section of this website called "The Physics of Sailing". I will give an extract from it.
There seems to have been no technology transfer from aerodynamics to sailing to solve the problem of finding a way of resisting the transverse force generated by a sailing rig with a minimum of penalty in drag. I found an author who claimed that figure 13 showed the evolutionary sequence from the deep-hulled Brixham trawler with integral rudder through blended keels with integral rudder to a separate fin and rudder. When I was thinking about this some time ago I altered the diagram to replace the blended keels with a fins with inset rudders to give figure 14. It is then obvious that the keel in the middle at the start of the sequence was well placed to balance the boat but not for steering it and that the movement aft improved the steering but upset the balance of the boat. It was a dead end and led to the final change which was to separate these two requirements and fit a rigid keel to balance the boat and a rudder to control the keel and to steer.
People will want to model versions of the first three designs in figure 13 and all three are just steps on the way to what was really needed. I do not know how these designs handled but they must have been unsatisfactory in some way for the evolution to continue. Presumably a model built to scale will also be unsatisfactory. I can see no cut-and-dried way to add an aerodynamic keel without modifying the hull but modellers do add keels to the scale hulls because of the stability problem.
It is hard to think how these first three boats were sailed but presumably, like ships such as the Cutty Sark, they were steered partly with the rudder and partly with the sails. I do not think that this is an option for scale modellers
The text above is dependent on being able to narrow down the position of the centre of pressure on a sail relative to the centre of area. I need to expand on that and have done so in this appendix.
We need to find out what we can about how the distance between the centre of area of the sailing rig and the centre of pressure of the sailing rig varies with the size and the number of sails and we can make use of the same information as we used for the fin.
We know from aeromodelling experience that the centre of pressure of a parallel chord wing is about 10% of the chord in front of the centre of area. Somehow we have to apply this to a sail that is likely to be triangular. That is not very convenient but we can get what we need by considering sails that are rectangular.
Let me start with a square that could represent a sail with it's luff on the left hand side. We know that the centre of area is in the middle at the intersection of the diagonals and for this exercise we could suppose that the net force on the sail acts at the centre of pressure at about 40% back from the leading edge. So the centre of pressure is about 10% of the width of the sail in front of the centre of area. I have laid this out diagrammatically in figure 15 where the square sail has a side of 10 units of length. Using the 10% idea again the distance between the centre of area and the centre of pressure is 1 unit.
Now we can consider the case of a sail of the same area but rectangular with sides of 20 units and 5 units with the long sides being the luff and leech. The outcome is shown diagrammatically in figure 16. The consequence for us is that the distance between the centre of area and the centre of pressure is now 0.5 units. So the aspect ratio of the sails affects the distance between the centre of area and the centre of pressure with the distance falling as the aspect ratio increases.
This immediately shows another problem faced by designers of Marblehead yachts. There it is normal to use swing rigs with very high aspect ratios for the top and working suits. It is normal to have three more rigs for heavy weather and the make the rigs of progressively lower aspect ratio. Inevitably the heavy weather suits will have their centres of pressure progressively further aft and, to correct for this, a second mast socket nearer to the bow is needed to avoid overworking the rudder.
Sailing rigs are mostly made up of two or more sails all with different functions in the overall scheme of things. We need know how dividing the area between several sails affects the distance between the centre of area and the centre of pressure. In figure 17 I have split the original square sail into two sails each 10 units by 5 units and we have a centre of area and a centre of pressure for each sail as shown in figure 17. If this is regarded as a sail plan so that this is where the sails would be on the boat it is easy see that the centre of area for the sail plan would be at the junction of the two sails and a simple calculation shows that the centre of pressure of the rig will still be 0.5 units in front of the centre of area. The effect of using two equal sails instead of one is to halve the distance between the centre of area and the centre of pressure.
Typically, in the days before swing rigs, the Bermuda rig on a Marblehead split the area of the sails 60%/40%. We can use this to split our basic square of sail into unequal areas in figure 18.
The centre of area of the rig is still at 5 units. The centre of pressure of the small sail is at 0.4 units in front of the centre of area and the distance is 0.6 for the large sail. By calculation the centre of pressure of the rig would be 0.52 units in front of centre of area of the rig.
These few calculations show that every change from the basic square that has the greatest distance between the centre of area and the centre of pressure reduces that distance by a significant amount. It seems to me that the most important outcome for us comes from figure 18. It is that if you know the distance between the centre of area and the centre of pressure of the largest sail the distance between the centre of area and the centre of pressure for the whole rig will be shorter by an amount that depends on the number of sails in the rig.
In the beginning I pointed out that we need to know how this length changes with size and numbers of sails for triangular sails. I ran the same exercise for triangular sails with the same result. So it would not be unreasonable to place the centre of pressure of the whole rig at a point that is 10% of the width at the centre of area of the largest sail in front of the centre of area of the whole rig.
 This article is not for the designers of full-size yachts. I know nothing about the way that they go about design.