center of flotation calculation and implications?

OK now...
to my knowledge sailarea and rig a boat can carry and will be designed for are calculated on the basis of RM at 30° heel and bft 8 wind forces...

in your example, mike, of an open 60 with an 15 and 5 foot keel RM will be considerably different and lower for the shallower keel although having the same displacement... therefore the sailarea should be reduced automatically...
i know a lot of designs which have 2 or more different keels/drafts one could choose and almost for every single design it is strongly recommended and designed as well that one should restore to the lower rig and lesser sailarea if a shallow keel variant is wished...
everything else would be irresponsible...

if you are fitting an open 60 with a 3 times shallower keel, same ballsat and displacement and still maintain full canvas - you should - sorry for that harsh words - receive treatment...
no skipper/owner should be allowed to do such stupid thing!
i think that the example you brought up is somewhat moot... sorry - but it is the way i think about it...

edit:
usually if the designer provides a version with a shallower keel - either the rig is smaller hence less sailarea or the ballast goes up (keeping RM the same) hence more displacement and both of those figures are represented in the S# calculations...
if i should meet any designer telling me "you can go for the shallower keel with the same ballast and still sail the tall rig" i am looking for the door already...
i really do not see your points mike...
sure enough - and that was discussed here in this thread earlier already - there are various factors more when it comes to performance prediction but as a general guideline i appreciate the proposed S#...

 

Once again: These ratios and numbers are for comparing boats of similar design. An Open 60 with a 15 foot keel is not similar to one with a 5 foot keel.

Yes, manufacturers could easily supply these data, but most often they don't/won't.

 

I think part of Mike's point is the S# may be misleading since it can assign the same value to boats of vastly different performance potentials. Since the S# was presented as a means to evaluate performance, what Mike is saying makes sense to me.

 

Cap'n V: SA calculations are fine, but there's multihulls and foilers to consider
(and the Bruce foil fraternity may be let out of the sanitarium any day now)

 

Mike, I see your point. By the same token, you could have an Open 60 with the deep keel, and a rig that has five very short masts but with the same sail area as compared to a comparable sister. But the center of effort of the sail plan would be much lower, the rig would be much less efficient, and the heeling moment would be a lot less. It would be a lot more stable by comparison. It would not perform well in light air, but may perform better in heavy air.

But both of these examples are extremes, and also very unlikely. One certainly has to take the ratios in context of the boat--of course you need some specific knowledge of the boat in order to interpret the results, and the first thing you would see with these two examples that you and I note is that they would be really weird boats. The S#, I think, is not meant to be a perfect, fool-proof predictor of performance for all configurations of all designs in all conditions. And it is not a rating or handicapping formula, guaranteed for all classes of boats. It is merely an indicator of possible performance of reasonably designed boats in reasonable conditions, nothing more. It gives some context for the plot of SA/D versus DLR so the one can perhaps gain some perspective in performance.

I am not mathematician enough to figure out how you would incorporate a stability factor into the S# equation and still make it work. And what kind of stability factor would be appropriate? Righting arm, heeling arm, areas under the stability curve, range of stability??? If you take a lesson from STIX number, supposedly a measure of seaworthiness, you have an equation that has both a displacement/length factor and a wind moment factor in it, and then on top of that, you have stability factors in the form of a Beam displacement factor, a knockdown recovery factor, an inversion recovery factor, and a dynamic stability factor. It finishes with a downloading factor. So, maybe pick one of these, and then, would you be able to figure out a result that would keep the S# on a scale of 1 to 10, as is its original intent? I think the S# would become a messy calculation suitable only for designers and builders with intimate knowledge of the design at hand. The average Jane or John Doe on the street could not calculate the S# for themselves which is another benefit that originator Peter Brooks had in mind, because stability information is generally hard to come by (although it needn't be, I agree.)

Rather than try to make one number do many things, what is far more reasonable to my mind is to use the S# in conjunction with other numbers and ratios that we already have. For example, in our weird boats above, two boats may have favorable S#s, but their idiosyncracies will show up in other ratios, STIX number for example, or simply by prima facia evidence that the boats just look funny. That will likely be enough to raise a flag that there is something wrong with the design that would require further investigation, and that S# would give a suspicious result.

As for Dellenbaugh Angle, it does derive from the GZ curve, as you will see in the derivation that I will show.

Eric

 

IMO, the S# can not assign any values to anything. The values are assigned by people using the S# formula. And if these people assign values to "boats of vastly different performance potentials" in order to compare these boats, then they have misunderstood something!

 

sure...
but who is going to compare predicted performance between multis, foilers and monos by means of the S#... ?

and yes - i was focused with my statement on ballasted monos only... i am a little narrow in my view there because those are the vessels i am looking for...
nobody is taking a foiler on a extended, ocean passage making cruise with his family onboard - do they?
and i am somewhat biased when it comes to multis...

 

Soren, Vimes


The original premise :

was not to compare similar vessels that correlate on other aspects outside the formulae. It was as a global scale.

The example I gave was simply illustrative of why I thought that premise was not met.
It would be nice to see other factors in that equation, a max allowable windward sail area could be then sensibly indicated by GZ. In other words incorporate a Dellenbaugh angle factor. Then an over canvassed vessel would have it's allowable input sail area cut down to something more sensible.

Anyway it looks like the S# is just another ratio to consider and not a standalone indicator, so it can rest at that:

That is what I was thinking too

 

Dellenbaugh Angle

Up to now, the discussions of the various design ratios all had to do with hull form and displacement. Today, we move a little deeper into naval architecture with a topic that touches on stability—Dellenbaugh Angle for sailing yachts. I suppose that a comprehensive discussion of stability should precede this, but stability is a very broad topic, and it would take several weeks to cover it completely. Instead, I refer you to the basic design texts such as Kinney’s Skene’s Elements of Yacht Design or Larsson/Eliasson’s Principles of Yacht Design for a thorough review. Probably one of the best texts that I have read on sailing yacht stability is “Chapter IV, Stability,” in Pierre Gutelle’s The Design of Sailing Yachts. Gutelle’s book is not the best overall, but the stability chapter is very good. Going forward from here, you should have a good understanding of stability and the concept of metacentric height, GM.

So now, Dellenbaugh Angle. First of all, who was Dellenbaugh? Dellenbaugh was Frederick Samuel Dellenbaugh, Jr., a professor of Electrical Engineering at the Massachusetts Institute of Technology (MIT), beginning as an assistant professor in the early 1920s. He was also the coach of the 1923 men’s heavyweight varsity crew team. At that time, MIT also had a prestigious department of Naval Architecture and Marine Engineering. In 1921, Dellenbaugh’s master’s degree thesis was on analyzing harmonic curves and plots by the use of mechanical machines like planimeters and integrators which we use in naval architecture. He even invented his own electric analyzing machine for this work. Interestingly, the righting arm stability curve of a floating vessel is a harmonic curve, and buried in the thesis (if one were to read it, which I did) you can see where Dellenbaugh Angle eventually came from, some of which I show below.

Frederick Dellenbaugh developed the Dellenbaugh Angle for sailing yacht analysis in the 1930s. How or why the calculation actually came about, I don’t know, but it is fairly simple as you’ll see below. It is known that Dellenbaugh had some correspondence with Olin Stephens of Sparkman & Stephens (S&S) in the 1930s, which would have been early in Olin’s career as a yacht designer. If you Google “Dellenbaugh Angle” on the Internet, all references point back to Francis Kinney’s version of Skene’s Elements of Yacht Design which was first published in 1962, and then later revised in 1973 (the version I have). Francis Kinney worked on and off for S&S from after WWII until the 1970s. So, barring any other facts that may be known by others, I would say that Dellenbaugh Angle came about during this association between Olin Stephens and Frederick Dellenbaugh, Jr.

Just to fill in the picture (because I like the history so much, and it gives some context), Frederick Jr’s. father was Frederick S. Dellenbaugh, Sr., who was a renowned artist, photographer, explorer and traveler. He participated in the second Powell expedition on the Colorado River in the 1870s, and in the Harriman maritime expedition of the coast of Alaska in the 1899. Frederick Jr’s. son was Warren G. Dellenbaugh, who was a principal in a company called US Yacht which was instrumental in the development of O’Day Yachts in the US, a well-known builder of fiberglass sailboats. Warren’s sons, and Frederick Jr’s. grandsons, are world-renowned sailing experts Brad and David Dellenbaugh.

Here is the equation for Dellenbaugh Angle (DA):

DA = 57.3 x Sail Area x Heeling Arm x 1.0/(GM x Displacement)

Where:
Sail area is in square feet.
Heeling arm is in feet and is the distance between the center of effort of the sail plan and the center of the lateral plane of the underwater profile of the hull.
GM is the metacentric height in feet.
Displacement is in pounds.

Consistent metric units will give you the same result, with the lengths in meters, area in square meters, and displacement in kilogram force. The number 1.0 (pounds per square foot) in the numerator must change to 4.883 (Kilogram force per square meter).

This is simply the heeling moment divided by the righting moment resulting in an angle of heel when the wind pressure is one pound per square foot (or equivalent metric measure). This may be hard to see, so let’s look further. You should recall from your study of stability that:

Heeling Moment = Sail Area x Heeling Arm x Wind Pressure (1 lb/sq.ft. in this case)

And:

Righting Moment (at any angle of heel) = Displacement x Righting Arm

We know that Righting Arm = GZ = GM x Sin θ

So, Righting Moment = Displacement x GM x Sin θ

Refer to the first figure, DA Chart 01, which shows the full righting arm curve for my Globetrotter 45 design, Eagle, which favored very well in the Westlawn Yacht Design competition in 2007 (it placed 7th out of 10 in the official judging, but it won the popular vote by a wide margin). At small angles of heel, you can see that the value of righting arm, GZ (black line), closely approximates its own slope at zero degrees of heel (green line).

The slope of the GZ curve = GZ/θ = GM x Sin θ/θ

Since Sin θ approaches θ as θ approaches zero, the slope of the curve at the origin is the metacentric height. That is, if the righting arm continued to increase at the same rate as at the origin, along the green line, it would be equal to GM at a heel angle of one radian, 57.3º. Look at DA Chart 02 which is a close-up of the origin of Chart 01. Notice that the black GZ line is close to but not exactly on the green slope line. At about 12º heel, the black line departs significantly from the green slope line, and this is because as the boat continues to heel, the shape of the waterplane gets narrower, so its moment of inertia is less, and as a result the metacenter slides down closer to the waterplane, and so GM is slightly smaller. As GM goes smaller, GZ goes not increase as fast, and so the black GZ curve starts to bend away from its zero-degree slope.

However, at small angles of heel, and without doing a full righting arm curve calculation, we can approximate the righting moment at 1º of heel:

Righting Moment at 1º of heel = Displacement x GM/57.3

If we divide the Heeling Moment at wind pressure of 1.0 pound per square foot by the Righting Moment at 1º of heel, we will get the approximate heel angle at which the righting moment of the boat equalizes the heeling moment. This will be the estimate of heel angle along the green slope line, the Dellenbaugh Angle:

DA = Heeling Moment/Righting Moment at 1º

Substituting for Heeling Moment and Righting Moment at 1º of heel:

DA = Sail Area x Heeling Arm x 1.0/(GM x Displacement/57.3)

Straightening out the math signs:

DA = 57.3 x Sail Area x Heeling Arm x 1.0/(GM x Displacement)

This is the equation for Dellenbaugh Angle that we started with. Obviously, the smaller the DA, the stiffer the boat is. Stiffer boats will generally sail better—be faster and point higher—than tender boats.

That is, Dellenbaugh Angle is the estimate of how much the boat will heel over in moderate conditions. One pound per square foot of wind pressure is equivalent to a wind speed of about 16 miles per hour, Beaufort Force 4. If you look at Skene’s Elements of Yacht Design, page 297 (8th edition, Kinney), you’ll see a chart of wind pressure versus wind speed in miles per hour, which is simply a plot of what Kinney calls Martin’s Formula, P = 0.004 x V^2. The student is encouraged to prove the validity of this equation. The 0.004 is the result of the density of air and the consistency of units using a wind speed in miles per hour against a sail area of one square foot.

Dellenbaugh Angle is quite accurate at small angles of heel, where the green slope line and the black GZ line come very close together. DA is less accurate at higher angles of heel where these lines start to separate.

Dellenbaugh Angle is quite useful for estimating the stability of a boat design early in the design process, before you have created the 3D hullform when you can do a full stability curve calculation. For example, when I was designing the Scandinavian Cruiser 40 in 2008, the client and I were going through various permutations of the sail plan as the overall arrangement of the boat was being worked out. I was working in AutoCad with my intended hull shape, but this was before I had even started on a 3D hull model. I made some estimates of the stability of the boat based on the drawings and weights up to that time, and I used Dellenbaugh Angle calculations to work out the heel angle of various sail combinations and wind speeds. This helped us focus on the desired sizes and arrangement of the sails. See the sample plot below, SC40 Heel Angle.

In Skene's Elements of Yacht Design, Kinney shows a table and a chart of Dellenbaugh Angles for various types of craft, and he separates these out by length and whether they are centerboarders or keel boats. The chart was first published in Kinney’s 1962 edition, which he then updated 10 years later. The caption to the chart states, “Today (1972) new keel sailboats are 25% stiffer than shown on this chart.” I dare say that this trend has probably continued in general, that boats with the current crop of keel designs are considerably stiffer than they were back in 1972 (now, 38 years later).

Well, that was a pretty involved topic, and I hope everyone understood it.

Questions?

Eric

 

Eric, how is the Dellenbaugh Angle affected by anti-heel devices such as Bruce foils, movable keels and/or ballast?

 

Thanks Eric,
There will be questions I think, some questions will be answered after I ponder this a bit. The rest will benefit from seasoning.

 

It's not. Not directly anyway. The only variables in the formula are metacentric height, sail area, and heeling arm. Nothing else effects the Dellenbaugh Angle. Changing the vertical position of ballast will change the center of gravity and will change GM. Moving ballast athwartships kind of throws everything out of whack with regards to the DA formula. The DA was much more useful in the 1930's through the 1960's when just about all cruiser/racers had similar proportions, and similarly shaped righting arm curves. Keep in mind it was a lot of work to generate a righting arm curve before everybody had their own computer. Using the DA as a benchmark was and is a relatively simple task.

I second Eric's suggestion to study Larsson/Eliasson’s Principles of Yacht Design and other sources for the fundamentals of static transverse stability.

 

Paul answered that perfectly, thanks!

Eric

 

We should all remember that these ratios are from before the arrival of PCs - from the days when all calculations were made using a slide rule.
Today, we have PCs and very sophisticated VPPs which -with a little bit of luck and some fiddling - can give us results that are a little bit more precise than the primitive ratios.
This weekend, I found this quote in the danish engineering journal The Engineer:

It is better to be vaguely right than exactly wrong
(Carveth Read: Logic: Deductive and Inductive, 1898)

That's why these ratios still have a role to play in yacht design!

 

Søren, you have an excellent point. My career spanned the time from when computers were a vague rumour to the one on every desk period. I was quite happy, and entirely accurate for the purpose, working things out on a slide rule, but when the computer arrived everything had to be exact. Particularly annoying was completing an important estimate of several million dolalrs, consisting entire of informed and not so informed guesses added together, highly unlikely to be within 10% at best, and then having it delayed by an extensive check because there was an error of a few pennies in the result.