# Once again on righting moment versus heeling moment

Discussion in 'Sailboats' started by laukejas, May 14, 2014.

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### laukejasSenior Member

Hi fellow skippers,

I'm working on a 4.2 monohull sailboat for 2 people. Balanced lug rig. 206kg displacement.

I'm trying to determine sail shape and size, was well as reefed sail area, so that I can know what the balance of the boat will be at various wind speeds.

I believe I understand the righting moment calculation. In my case, it goes as follows:

GZ (waterline-paralel distance between buoyancy center and center of gravity) at 5° heel angle = .32m.
Displacement = 206kg.

RM = GZ * Displacement * gravitational acceleration = .32m*206kg*9.81m/s^2= 651Nm

Now, for the heeling moment of the sails:

Sail area = 9m^2
Heel angle = 5°
Wind speed = 7m/s
Air density = 1.225kg/m^3
Height of Center of Effort = 2.56m

Wind pressure = 1/2 * Air density * Wind speed^2 = 1/2*1.225kg/m^3*(7m/s)^2 = 30.01
Wind force = Wind pressure * Sail area = 30.01 * 9m^2 = 270.11
Heeling moment = Wind force * Height of Center of Effort = *270.11 * 2.56m = 691.5Nm
Adjustment for heel angle = Heeling moment * cos(Heel angle) = 691.5 * cos(5°) = 688.8Nm

Since I have never ever done this before, I am not sure if my calculations are correct.

Could someone please check and correct? Are these formulas accurate enough?

P.S. what are traditional windspeeds for boat of similar size to put in 1st and 2nd reef?

P.P.S. One of the things I'm not sure is if wind speed should be measured in m/s, km/h, or knots...

Last edited: May 15, 2014
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### SkyakSenior Member

Hi laukejas,
while the result looks good to me I can't follow how you found it. Gz is typically used only on keel boats. To be honest I have never seen a really good calculation for dingy stability. Mostly I see the calculation of the weight of the crew times the distance from the center (CB) to the position they would sit on the deck equal to the force on the sail times the distance from the center of effort of the sail to the center of effort of the dagger board. The force on the sail per height is equal to the reynolds number times density times the coefficient of lift. CL of 1 is about right so I think that worked out for you.

Wind speed of the first reef varies based on the local conditions. If local wind tends to be light you will want a large sail and reef early. I think we should create a spreadsheet to consider how well the boat will sail with 2 crew and light air and solo in heavy wind. If you can cover both you have a good rig.

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### laukejasSenior Member

Skyak, thank you for your reply. You devotedly help me out again It's good to be back.

I made a very complex model of this sailboat in Solidworks (CAD software). I included model of two persons in that sailboat, and set their position more or less to real hiking situation (sitting on the gunwales, leaned back).
This software is able to calculate Center of Mass for the boat AND the crew together.

So, I just drew two lines - one from center of mass, one from center of buoyancy at 5° heel, both parallel to each other and perpendicular to waterline. Distance between these lines is GM.

I actually expressed formulas in a way that I edit the sailboat model (for example, shift crew position, change heel angle, change sail shape or reef positions, etc.), and Solidworks tells me what wind speeds can each configuration carry. Very automated.

Right now, it says (at 5° heel):
Full sail: 7.06m/s
Reef 1: 10.03m/s
Reef 2: 13.61m/s

This doesn't look too good, as I want boat to be manageable in 20+ m/s gusts. Reef 2 now gives sail area of 2.96m^2 and it's CE is 204.67cm above waterline.

Maybe I should consider a stormsail, I don't know. Never that saw on dinghies, though. Maybe another reef point?

I'm attaching a screenshot of the model, just for reference. Model is still incomplete, many parts missing, but it's enough for rough calculations of weight and displacement, I guess.

http://i.imgur.com/qNIah7U.jpg

Can you explain that a bit, please? I highlighted the part which is completely beyond me right now...

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### philSweetSenior Member

Accuracy isn't the issue here. This just isn't how to go about it.

One old rule of thumb was to estimate the sideforce of the sail at 1 pound per sq foot of plain sail. Use the vertical distance between the sail's centroid and the centroid of the fin. Ignore the hull and extend the fin's leading and trailing edges straight up to the surface in order to estimate the vertical center of lateral resistance as the centroid of the extended fin.

Once you have this, there are all kinds of rules-of-thumb and finagling factors that can be applied to specific types of boats, rigs, and sailing habits. The ability to hold this pressure generally indicates a reef needs to be tucked in the 18-20 knot TWS range, so it might be a bit high for a pond dinghy. Baseline of RM @ 30 = SF @ 1PSF.

The following chart came from the Plyboats booklet. I have not received any reply from repeated requests (over the past 8 years) to reproduce charts from this booklet. I believe they are properly in the public domain now. You can also check the data on http://sailboatdata.com/, but I have found a lot of errors on that site, so if you key in on one or two comparables, I would advise you to get the data from the mfg.

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### laukejasSenior Member

Well, but this doesn't include wind speed. I take your meaning that this is general approximation for finding sail area that would be small enough to sail up to 15 knots, and then determine each reef by reducing 25-30% of sail area, right? I believe I saw something like this before.

This rule of thumb seems very approximate for me. Why is my described method (presumably more accurate) not suited here?

Why is hull ignored in calculations?

By the way, according to the chart you provided, sail area for my boat should be around 10.2m^2, considerably larger than current (8.81m^2).

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### laukejasSenior Member

I believe I managed to think this out. However, I see three problems:
1) Sail area isn't included in formula;
2) Weight of the boat itself is also not included - after all, it weights 40 kg, and affects CoG considerably;
3) How can I find out Reynolds number?

If I got it correctly, I converted formula to metric system, and finally got 392Nm of heeling force for a 8.81m^2 sail and 3.05 distance between CE and CLR of the fin. Is this about right?

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### SkyakSenior Member

laukejas,

the righting moment you got from CAD is excellent and with that I would say your calculation makes sense and worked with the possible exception of the distance between the sail CE and dagger board CE. The reason the hull vertical area is not counted is that in a fast dingy like yours the hull is shallow and rounded and will not and should not produce much lateral force at the low leeway the daggerboard dictates. If solidworks counts vertical hull area it is because it assumes a deep keelboat.

I am surprised that solidworks does any aerodynamic or hydrodynamic calculations. What version are you working in? I would also say you have developed some CAD talent. If you are still short money for the project maybe you could freelance CAD work.

It has been pointed out to me that I incorrectly cited Reynolds number. What I was trying to say is that your calculation worked because a coefficient of lift equal to one is about right so your Bernoulli pressure times area is equivalent.

I am not sure if you know Phil but he is a greater authority than I am. The way he describes determining dingy sail area is the way I have seen it done most, rough estimates and compare to past examples. For every improvement in performance calculations there are ten ways it could be misleading. That said, I think there is value in more precise modeling of what is going on but I would not delay your project one second -the old ways are good enough to build to.

About righting in a gust; all of these estimates are for the sail at an angle of attack close to stall around CL max. In operation you will not cleat the sheet. The crew will hold it most of the time and play it out when a gust heels you over. I have not seen any calculation for ultimate wind force (force in a strong wind with little or no force on the sheet) but I do know that the best thing you can do is add battens to stop the sail from flapping and causing drag -more like a junk rig than a lug.

There are also 'soft wing' rigs that are like junk rigs with a wing shape forward around the mast. I have been thinking about prototyping one of these for my laser. Another rig you might be interested in is Matt L's Paradox -a balanced lug that roller furls onto the boom providing easy continuous reefing.

So good work and though I don't want to slow you down with 'paralysis by analysis' I find it hard to resist doing the math when you have such an excellent model. This forum has been pretty dull lately and I am interested in what Phil has learned from the aerodynamics class we are taking.

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### laukejasSenior Member

Well you went ahead of me! This was my next question, how can I calculate lateral area if hull is shallow. I assumed that if hull has little or no slope ( U - shaped ), then it's lateral area should be calculated at 100%. If it were V shaped with 45° slope, only 50% of it's lateral area should be calculated. If it were flat-bottomed and very shallow - near 0%.
This bugged me for a very long time. I guess with dinghy as shallow as mine, hull lateral area can be excluded from calculations, as you suggest. It'll make things easier, and in practice, if hull does provide some additional lateral resistance - why not?

Speaking of which, how do I size fin proportionally? Lars Larsson and Rolf E Eliasson suggest that it should be 3.5% of total sail area. However, this is for yachts (like YD-40), not for dinghies. Since hull provides little lateral resistance in my case, I guess fin should be larger - but by how much?
I really don't want to under-size it. Lithuanian lake winds tend to be treacherous, for you can expect dead calm after a 30kt gusts a minute ago. And I know how difficult is reaching in glassy waters when you have to tack two times a minute due to narrow passages.

My point is, I want this boat to perform reasonably in 1-3 knot winds. Since in low speeds fin will act less like wing and more like a keel, so I guess I just need it big.
Question is, how big? Right now my fin extends around 90cm below waterline, and weights 2.6kg. It is exactly 3.5% of the sail area. How much meat do I add?

Well, it doesn't. It has nothing on this. And it works slow with Top-Down designs. However, it has equations. It can measure weight, area, distance, center of gravity, and so on, and outputs these in numbers. Then, it is only a matter of dedication to input all the formulas. Right now, I have around 20 equations which I took mainly from the "Principles of sailboat design", then ~10 other books, wikipedia, this forum, common sense, and then basically programmed SolidWorks into sailboat stability calculator.

This is how it looks. Took me a week to program (mostly to learn). Sorry it is in Lithuanian - I prefer to work in my native language (it also encourages me to learn nautical terms). But you can get the idea of how scary it looks:

For example, numbers 7 to 9 are lead of CE ahead of CLR in cm for full sail and reefs 1&2, 10-12 - same in percent of total waterline. 15-17 are areas for full sail and reefs. 20 is aspect ratio of the sail, 21 is lateral area of fin, 22 - sail area/lateral fin area ratio in percent, 23 is righting moment, 24-26 are maximum wind speeds for full sail and each reef.

Highly complex, very accurate. I just have to adjust the model - for example, shift crew position, add a new part, re-shape sail, change hypothetical heel angle (transverse, longitudinal, or both), draft, and these numbers update, so I can quickly see how it will affect boat performance.

I doubt anyone would pay money for my beginner skill. But thank you for the compliment

Well, right now, I have time. I am now deep into my studies (music), I finish Masters next year, so I plan to build this boat next summer, after final exams. It means I have a year to finish this project. Most things are done, but there are still 30+ items on my list to input in this model.

That being said, if I have time, why not make it a bit more accurate?

That's okay! I find it also hard to resist. There is time, so I want to perfect my boat design. There just so many things to make work together. Requirements are steep: folding sailboat for 2 persons, \$700 budget, folding so that no part is longer than 2 meters, car-top-able (so it must weight under 40kg), fast enough to assemble/disassemble within an hour (no screwdrivers or pliers included, no fasteners that could be lost), sailable (reaching ability) in 1-25kt wind speed range, easy on the maintenance, sleeping space for 2 persons (not very comfortable, but possible with current design). It shouldn't be a racer, something between cruiser and a racer, more towards cruiser. However, there is one quality I highly value: it is minimum True Wind Angle when reaching. Not VMG, but TWA. This is because I'll often sail in narrow waters, meaning that if I have high TWA, I'll have to tack very often, and each tack is loss of speed, loss of ground. This is why it would be far better to be able to reach high rather than fast in such conditions. In more open waters, naturally, VMG is more important, but, as I said, I expect a lot of maneuvering.

Most of these requirements seem reasonable with current design. I make regular trips to my local hardware store, look what parts are cheapest, measure them, go back home, open up Solidworks, try to combine them, adjust requirements, go back to hardware store again... Very tiresome work, heavy on the head.

But if I manage to actually design and build it, it'll be really something.

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### SkyakSenior Member

Well if you have the time and interest in doing the calculations I am all for it!

Considering the hull as a foil for lateral force on the water moving at say 4 kn, if you angle the hull 3 degrees you will feel a kg or two of lateral force and about a kg of drag added -terrible lift/drag. If you do the same exercise with the daggerboard down you will feel 100+kg of lateral force and mid to high double digit lift/drag. So despite the fact that the the hull may have equal or greater lateral area than the DB it is so poor as a lateral foil that you should give it no weight in calculating righting moment. If you averaged it with the DB area it would move the CE much higher as if the hull lateral area worked as hard as the DB area -wrong.

The reason the hull is so poor laterally is that it has super low aspect ratio. The free end of a foil has zero lift. All the pressure it creates just pulls fluid around the end from the high pressure side to the low. As we move along a foil away from the end and toward the middle the pressure increases toward the theoretical 'infinite span' value. This is why performance focused designers are so obsessed with aspect ratio. High AR gives the highest performance/area. The one exception to high AR is sailing straight downwind (where sail drag would be a benefit) AR of 1 would be best. And before I leave span-wise load let me say that at BEST the load distribution will be elliptical.

So the 2 dimensional lift per unit length of an infinite foil is half*density*velocity^2*cord*coeficient of lift. If the average Cl=1 then the lift per length times length of the sail is the same as your Bernoulli pressure times area. I said this in my last post but I bring it up again because it is the minimum theory you have to consider to relate foil size, force, and fluid speed. With this in mind we can go back and figure what wind speed the standard practice of assuming one pound/sf (4.88kg/m^2) sail area is based on. I get 8.8m/s with the cl=1 and 10.5m/s if the cl was only 0.7 (possibly more realistic average for AR<2. So the 1 lb/sf on the sail calculation prepares you for some pretty high wind speeds with the sail sheeted in.

You asked about dagger board size of 3.5% of the sail area -that's in the ballpark. It looks like more than enough in the picture. If you compare to other dingy designs I think you will find a surprising variety of ratios because it depends on the relationship between air speed and water speed according to the calculation above. On very fast catamarans and winged skiffs you will find small slender foils and huge sails because they operate at double digit water speeds. I don't think your daggerboard needs to be any bigger. My recommendation would be to consider moving it forward so the front of the DB trunk is on the same frame as the mast -then make it a bit smaller and make the rudder bigger. My reasoning is that the mast and the daggerboard trunk have the largest forces so I want them attached. This moves the DB forward of the 'optimal' position which will result in more weather helm. This is no problem with the right size rudder and will actually feel better with more communication of force and stall.

If you want to read about designing a high performance dingy daggerboard here is an article about doing a 505 (two man 5.05m).

The estimate of force and wind speed also puts your other performance desires into perspective. 1 knot wind will not push the boat to any reasonable speed. It just isn't wind and you can't put up enough sail to make up for it's lack of energy -you are better off paddling. 3 knots is something to work with -with enough speed for control and some sense of direction.

25 knots of wind is well past "small craft advisory". I don't know what waters you will be in but 25 kn makes big waves fast and rips their heads off. I think 25 Kn wind is good for calculating boat strength but if it ever happens you will be looking for a safe landing down wind. If you really want to sail in 25 kn you will need a self bailing cockpit and an unsinkable hull.

If you just need to handle 25kn gusts we should look at 'soft wing' designs because their super low drag allows you to just sheet out and stay upright. The soft wings are also great for upwind sailing. The soft wing is like a junk rig but the forward 1/3 of the sail has two sides that go on either side of the mast. I have been thinking about prototyping a small one for my laser to make it easy to rig and sail in strong winds.

Cudos on the programming! I still think it might be a profitable skill and I am wondering which version of solidworks you have? I think I have SW 2001 on one of my PCs.

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### laukejasSenior Member

Thanks for the info, Skyak. I really appreciate you helping me out on such detail. And I'm sorry, but due to my lack of experience and knowledge, I don't understand all of what you're saying. I have several questions...

You meant lateral area? I can't imagine how this relates to righting moment...

Are you talking about hull or sail here? I realize same principles apply, but just want to make sure. I know the basics about AR, I just hadn't realized that hull works like a foil too.

I'm sorry, but I really don't understand this. You're talking about relating sail area to wind speeds, right? The formula you provided, is it in metric or imperial? What units? And how the heck does one calculate Cl or the sail?

Please, take a look at this screenshot now:

http://static.dyp.im/QGZ46BC4zt/a5ccdb2904e0fa5907e0eef410c05ee7.JPG

The small blue points indicate stuff. Zoom in to see labels. On daggerboard, "Sverto SPC" stands for CLR. On sail, BC is center or effort, BC R1 and BC R2 are centers of effort for reef 1 and 2 (these three centers are almost vertical). To the right of BC there is ^SPC, which is vertical projection of lateral resistance (so that the distance between CLR and CE is easier to see).

This screenshot is done after I re-designed my model so that hull lateral area is not included in calculations, only daggerboard, as per your suggestion.

This moved CLR forward a lot. I had to shift the sail forward too, so to keep CE in front of CLR with decent lead. Right now, it is 6.1% of the waterline. However, with such sail position, mast is too short. I can't make it any longer due to design limitations. Also, I'm afraid the bad tack will be really bad.

If I were to move daggerboard even more forward, there is no way to have any kind of CE lead over CLR.

I use 2013 version... I looked it up, and sadly, I can't export it to your version. But I can provide all the information you require, including screenshots.

By the way, various sources suggest that lead should be anywhere from 5% to 30%, because when boat heels, CE actually moves backwards, creating weather helm. If one were to design boat with CE already behind CLR, weather helm would be unmanageable.
- This is not my thoughts, this is what I red on this forum and some other sources. But it seems logical to me. I also noticed that virtually all boats have CE in front of CLR, including Klepper Passat, the folding German boat on which my design is based.

I am sorry I ask so many questions in one post. I hope it's not too difficult to keep track of them all

P.S. Maybe I should include rudder in CLR area and center of area calculations? Then I could solve sail position/mast problem.

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Skyak-is this a misprint: ".... DB it is so poor as a lateral foil that you should give it no weight in calculating righting moment."?
Daggerboard area as it relates to the Center of Lateral Resistance and it's distance below the waterline would be used to calculate heeling moment. On a dinghy the distance from the Center of Effort of the sails to the Center of Lateral Resistance= the Heeling Arm. The force on the sails(pressure in pounds per sq.ft.) times the HA= Heeling Moment, right?

click for better view:

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### laukejasSenior Member

Doug, are you sure that the heeling arm is the distance between CE and CLR, not Center of Buoyancy? "Principles of yacht design" suggest the latter. It says nothing about CLR at this calculation.

Excerpt from the book:

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### DCockeySenior Member

Righting moment due to the interaction of buoyancy and the weight of the vessel is proportional to the horizontal distance between Center of Buoyancy and Center of Gravity, as shown in the figure in laukejas' post above.

Heeling moment due to the interaction of aerodynamic forces and hydrodynamic forces is typically assumed to be proportional to the vertical distance between "CE" and "CLR" as shown in the figure in Doug's post above. (Reality is more complicated but for most purposes this assumption is sufficient.)

If the Righting Moment = Heeling Moment (with sign reversed) then the net rolling moment will be zero and the boat will tend to stay at a constant heel angle.

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Yes,ask an NA or go to page 181 of Frank Bethwaites "High Performance Sailing".
Remember, this is "Heeling Moment" not righting moment and is specifically for a dinghy that is generally sailed close to flat. The CE of the sails pushes the boat to leeward and the combination of hull and board planform area(Center of Lateral Resistance) resists the leeward movement of the boat. The underwater area has the same force on it as the sails have, except opposite.
Note- in the illustration from p.181 of Bethwaites book you can see that the RM=HM.
And that the HA is from the CE of the sails to the CLR(vertical). On lightweight, fast dinghies just find the center of the board and discount the hull area(in my opinion) for this calculation.

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### laukejasSenior Member

Well, thank you for explanation. Dumb me. It makes sense now. I updated my model. I now get 7m/s maximum wind speed for a 8.13m^2 sail.

What about the lead I mentioned earlier? Currently, it is only 3.26% of the waterline (CE ahead of CLR). Is that enough? For reefed sail, do I need different lead?

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