# center of flotation calculation and implications?

Discussion in 'Boat Design' started by capt vimes, Jan 7, 2010.

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### Eric SponbergSenior Member

Thank you, Paul, I appreciate the sentiments. Exercises like this also help me keep my mind and understanding fresh. I have a lot of books strewn around my office floor at the moment, ready to check my facts.

Eric

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### DCockeySenior Member

I read "Principles of Naval Architecture" quite a number of years ago as an undergraduate though I was in mechanical and aerospace engineering, not naval architecture at the time. I did spend a year at Michigan getting a Masters in NA specializing in hydrodynamics and then decided to go elsewhere for further study in aerodynamics.

Yes, its a complex topic. The approach discussed of Ct = Cf + Cform + Cwave is the conventional naval architecture approach, and the basis of (virtually) all correlation of tow tank tests and use the results of those tests to predict real world performance. The potential for confusion arises when it's assumed that the actual physics of the water flow around the hull and the resulting drag and other forces are as simple as this approach would seem to indicate. There are higher level interactions which can limit the usefulness of any single set of "curves" and data. This is sometimes covered in more advanced studies of hydrodynamics. The ITTC line is not fundamental physics, rather it's a widely accepted, compromise empirical curve which is useful. The "Skin-Friction Formulas" in the Software forum has touched on this. http://www.boatdesign.net/forums/design-software/skin-friction-formulas-31280.html

None of this takes away from the general usefulness of the Cwb, Cb, Cp, SA/D, SA/WS, etc to compare boats of generally similar form. I have appreciated Eric's and others' contributions to this discussion.

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### Eric SponbergSenior Member

On A Scale Of One To Ten--the "s" Number (s#)

ON A SCALE OF ONE TO TEN—THE “S” NUMBER (S#)

Wouldn’t it be nice to rate the performance of all sailboats on a scale of one to ten?

Here’s the problem—we have different ways to rate a sailboat’s potential performance in the form of design ratios, handicap rules and ratings, and level ratings. In fact, rating systems have been around for centuries, dating back to England and the realm of Queen Elizabeth I—over 400 years. And in all that time, sailors and designers have continuously argued over what makes a boat go fast, and what should be measured and rated in order to allow disparate designs to compete on equal terms. VPP programs and CFD codes have tried to make performance ever more definable, but these tools require sophisticated programs and specialized people to run them. An alternative solution is to race in one-design boats, but, unfortunately, not everyone wants the same boat. On top of that, not everyone wants to race. Still, we want to be able to judge performance—we always want to know about performance.

A similar problem crops up with advertising hype—this or that boat is a racer/cruiser, cruiser/racer, racing machine, or simply just a dog that can’t get out of its own way. Who defines these things, and how is anyone supposed to make sense out of it all?

A rating number from 1 to 10 might simplify things for the average sailor and designer. What can we do with the information we already have without resorting to a consultant—some way that anyone can rate any boat on a scale from 1 to 10? Has anyone done this? Yes, someone has.

Back in the mid-1980s, I designed a “Boat-in-a-Box” sailboat—that is, a boat that could ship inside a standard 40’ shipping container—for a client in Texas, Mr. A. Peter Brooks. At the time, he and I both were also consulting for Cat Ketch Yachts Inc., the builder of the Herreshoff and Sparhawk cat ketches. Brooks was a retired business consultant and author, and he did some writing and marketing for the company. I designed all the carbon fiber free-standing masts for the boats. Brooks invented the idea for what he called the “S” Number (S#)—a single number between 1 and 10 which could rate the performance of all sailboat designs. This idea was published in Telltales, a southern Texas boating magazine, in April, 1988. I have never seen anything like it, before or since.

The concept is rather simple and is based on the Displacement/Length ratio (DLR) and Sail Area/Displacement ratio (SA/D), both of which we have discussed in the last few weeks. We know that DLR relates to drag—heavier displacement for a given length results in more drag, and boats with high DLRs are slower than boats with low DLRs. We also know that SA/D relates to power—more sail area for a given displacement results in more speed, and boats with high SA/Ds are faster than boats with low SA/Ds. We have also plotted SA/D versus DLR in a chart, and have seen how the spread of data points relates to boat performance. We use these same ratios—SA/D and DLR—to calculate S#, so we don’t need any new computer program to achieve our goal—just one new equation.

The equation for S# is an exponential and logarithmic function using DLR and SA/D as the primary variables. We already know how to calculate DLR and SA/D, and I am going to remove the slash “/” from SA/D so that it is less confusing in the S# equation— we’ll use the term “SAD.” Although the S# equation looks complex, it can be easily programmed into a calculator or a spreadsheet. Here it is:

S# = 3.972 x 10^[-DLR/526 + 0.691 x (log(SAD)-1)^0.8]

Brooks developted this equation with the assistance of Dr. Fred Young, at the time Dean of the College of Engineering at Lamar University in Beaumont, Texas. I spoke with Dr. Young by telephone some years ago just to make sure I understood the equation correctly, and he was very helpful.

Brooks collected a list of boat designs and their particulars from various published sources and calculated their S#s. Then he classified the boats according to the following categories:

· Lead Sled: S# = 1 to 2
· Cruiser: S# = 2 to 3
· Racer-Cruiser: S# = 3 to 5
· Racing Machine: S# = 5 to 10

The reasons for the ever-broadening scale of category names is simply a function of the logarithmic scale embodied in S#. This appears to be an asymptotic function. You can never reach the number 10, and you can never reach the number 1, both of which are the asymptotes.

Now we have a way to definitively categorize boats, not a wishy-washy, vague notion; we got a unique number for each and every design! I attach the spreadsheet that I used to calculate the SAD and DLR numbers in that chart I posted last week (S# Chart 01). Included in that spreadsheet is the S# calculation (pink column), and next to that is the category name for each boat. I sorted the data according to descending values of S# so that you can see how the categories play out. Also included in the spreadsheet is S# Chart 01 of SAD vs DLR for these boats. There are other charts there, too, which you can study or modify at your leisure.

The magazine sources for these sailboat designs are listed at the top of the data table and in the left-most column with the date of publication in the second column. They are all published data from advertisements and design reviews that I have collected over the years. As I review the magazines, I continually add data to this table. The original Telltales data that Brooks used and published in 1988 is included. I find it discouraging that in recent years the magazines have been slacking in publishing worthwhile design data on boat designs. Sometimes, it is difficult to get even the most basic of information—some small piece is frequently missing, and you don’t necessarily find it on the manufacturers websites. But we gather what we can.

Now here is where my contribution to S# comes in. Overlaying this chart of SAD vs. DLR, I have calculated and plotted the traces of constant S# so that you can see how they subdivide the boat population. I attach this chart as well (S# Chart 02). In the data table, I highlighted in yellow a few of my designs, and then have labelled them in the chart, just to give you some context.

So what do we see? S# Chart 02 can be interpreted as follows:

A boat that is very lightweight and has lots of sail area will have a low DLR and a high SAD. It has a high power-to-weight ratio, and so it will be very fast. Its S# will be between 5.0 and 10.0. It will be a “Racing Machine.”

On the other hand, a very heavy boat that has a small sail area will have a high DLR and a low SAD. It has a low power-to-weight ratio, and it is not going to be a very good performer. Therefore, its S# will be between 1.0 and 2.0. It will be a “Lead Sled.”

For S# values between 2.0 and 3.0, the boat will have a decent amount of volume to carry people and goods but won’t necessarily be a real hot-shot sailer. We can place these boats in the “Cruiser” range.

For S# values between 3.0 to 5.0, the boat will be in the middle ground between “Cruisers” and “Racing Machines”, so we can call them “Racer-Cruisers” (or “Cruiser-Racers” if you prefer.)

Therefore, the net result of the S# is a clear delineation of sailboat performance using a convenient scale from 1.0 to 10.0, and by this we give definitive meaning to typical descriptive names. In fact, Brooks claimed that the S# is a fairly reliable predictor of PHRF or IMS rating. For two boats of the same length, the one with a higher S# will be faster, will take less time to sail around a course, and therefore will have a lower rating. However, in the same article, this footnote appears: “Both Dr. Young and the author stress that the ‘S’ number is not a handicapping or rating system, but a guide to probable boat performance vs. other boats of comparable size.” I personally agree with that opinion.

Something else that is quite interesting is shown in the next chart (S# Chart 03) also included in the spreadsheet. I had the notion to divide SAD by DLR and plot that against S# and got a surprising result. All of the data forms a unique cluster in a very well-defined curve. These are two independent functions plotted against each other. Rarely in science do we see such a profound correlation of data. I am not absolutely sure of the ramifications of this, and maybe I am reading into it more than I should, but I would have expected a broader spread of data points in this chart. The relationship of the SAD/DLR ratio to S# is extremely solid as indicated by the cluster of points along its trend line. The equation for the trend line shown at the top of the chart is another way to approximate S# in a simpler cubic equation. Throughout the lower categories, S# follows the trend line almost exactly, and it is only in the Racing Machine category where there is some scatter away from the trend line. If we plot S# versus some simple dimension or factor such as LOA or Displacment, we see no discernable relationship to S# at all. But S# vs. SAD/DLR gives us a very unique view of sailing performance.

I am not mathematician enough to explain why this works as we see it. I have, however, on occasion, presented plots like S# Charts 02 and 03 to clients to review the performance they have in their current boats or are trying to achieve in a new design. It seems to give them a clearer understanding of what their current boat does or their new boat is going to do. S# is a way to be a little more scientific in layman’s terms. This gives us a better tool to clearly showcase performance without having to go to the model tank, measure resistance factors, plot results, correlate them to full scale, and then do VPPs on top of all that. A picture says a thousand words, and this seems to do a pretty good job.

You will also see in the data table and in the charts a calculation and plot of Ted Brewer’s Motion Comfort Ratio (MCR) plotted against S#. I will explain that in next week’s topic.

In the meantime, you are all free to use this database and spreadsheet as you please. You may add to it as you do your own designs or review the designs for others. You may change it around and expand it however you want. You may do other analyses and manipulate the data at your will. Send it to your friends, fellow designers and fellow sailors. Pass it around. Discuss it. Use it. The S# is for the public domain, and I hope it adds to our better understanding of sailboat performance. Time will tell.

Eric

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### mcollins07Senior Member

Thanks Eric. Very good stuff.

~ Michael

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### sorenfdkYacht Designer

No - this is not very good stuff. It is GREAT stuff!!!

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### TeddyDiverGollywobbler

Halibut 33 LOA33; LWL31; B10;D4,7; Disp14500; SA699; ketch; stripplank; DISPL/(.01LWL)^3 217,29; SA/VOL^2/3 18,81; S# 2,70; CRUISER; COMFORT RATIO 32,79; GREATER; (SA/Vol)/(D/L) 0,086560
Reliefing like navigating in a fog from EP with DR to where you want to find yourself
Thanks Eric! a great lesson!

7. ### Paul KotzebuePrevious Member

Eric,

Interesting stuff. I've been looking into the Square Meter Rule and have designed three hulls for the 40 sq meter (the boat I want) and 15 sq meter (the boat I can afford) classes. If I apply the S# to those boats I would expect:

1. The S# should be about the same for the boats in each class. I would not expect the S# to be refined enough to predict subtle performance differences.

2. The S#'s should put the square meters in the Racer-Cruiser category. Between 3 and 4 seems about right.

3. The S#'s for the 40 sq meters should be higher than the 15 sq meters. This is what I would expect based on trends in the Square Meter Rule.

Here is what I got:

15 Square Meter Class
Hull 1 14.64 134 3.22
Hull 2 15.82 154 3.14
Hull 3 18.40 162 3.39

40 Square Meter Class
Hull 1 19.49 144 3.81
Hull 2 17.04 121 3.83
Hull 3 15.25 103 3.81

All about what I would expect. Now if we can only get boat builders and designers to publish accurate and consistent displacement numbers.

8. ### Paul KotzebuePrevious Member

I think this tends to validate the S# approach as a means of comparing general types of boats. For example, all 12 meters should have about the same S#. The 12 meters with higher SAD's will have higher DLR's. Because the speed differences are so small between boats, the effects of higher SAD should be cancelled out by the higher DLR for the same rating, and should result in about the same S#. Same can be said for IOR one tonners. The 12 meters will cluster around one S# and the one tonners will cluster around another S#.

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### fredschmidtNaval Architect

Excelent job and idea. Is more an excelent pointer to help us.

Congratulations

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### capt vimesSenior Member

/signed - this is absolutely GREAT stuff!
thank's eric!

i was completely astounded to find an Ovni395/Alubat with a S# of 4.47 that high up in the charts...
its a hardchine, centerboard yacht... compared sometimes as a 'all-terrain-vehicle' of a sailing yacht and definetely not a RACER-cruiser... now thats something to think about...

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### Eric SponbergSenior Member

Thank you everyone for your nice comments. So far--24 hours later--S# seems to be getting a good reception.

Capt. Vimes, I guess the nice thing about S# is that it relates only to the DLR and SA/D ratios--nothing else. Hard chines, centerboards, etc. and anything else that could color our perceptions of a boat really don't enter into the performance evaluation that S# gives.

I did just go to the North Sea Maritime (Southend-on-Sea, Essex, UK) website to see the construction of the Alubats--fascinating! These are multichine aluminum hullforms that are built on jigs that hold all the pieces in place for assembly and welding--kind of like a mold for composite construction. This is the best way to manufacture production aluminum boats.

Eric

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### sorenfdkYacht Designer

Which is another reason to remember that all these ratios should only be used to compare yachts of similar size and type.

13. ### Paul KotzebuePrevious Member

If your result sounds too good to be true, it probably is.

Bear in mind it is common for published displacements to be lower than the actual sailing displacement. Also, make sure the sail area is the mainsail area without roach plus 100% foretriangle area.

If the boat in question is "definitely not a racer-cruiser", and your input numbers are accurate, and the resulting S# puts it in the racer-cruiser category -- you just invalidated the S#.

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### Eric SponbergSenior Member

I would turn that around. Remember, the S# definitions came from Brooks originally, without paying attention to the evolution of the descriptive terms prior to that. Prior, the terms were vague; now they are more clearly defined. I would say that the S# is very specific and it puts you in the named categories regardless of what has gone before. Rather than the readers change their perceptions, the builder instead may want to change his stated type to that which the S# declares. In the end, it really does not matter what you call it, because you can compare the S#s of different boats and still make a conclusion: the boat with the higher S# should be a better performer--in general.

Eric

15. ### Paul KotzebuePrevious Member

Eric,

What I meant to say is if the S# predicts a boat will have the performance of a "racer-cruiser" but it actually performs like a "cruiser" the S# is not valid. False predictions will invalidate the formula.

I am interested to know the derivation of the S# formula. Is the Telltales article available?

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