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Slogging to Windward
by Chuck Merrell
sloggingtowindward@hotmail.com 

February 2001

DESIGN ANALYSIS RATIOS
Part 2 - "Interpretation"
DESIGN ANALYSIS RATIOS
Part 2 - "Interpretation"
(be sure to read Part 1)

Like many others, I've been watching the current Vendee Globe "single-handed, round the world non-stop" sailboat race using Virtual Spectator with great interest. Today, January 31, 2001 is the 84th day of the race, and with only about 2700 miles to go, it looks like the winner will set a new course record of fewer than 100 days. At the moment, veteran sailor, Michel Desjoyeaux holds first place sailing PRB. However, the real story of this Vendee Globe is in second place in the person of 24 year-old, 5'2", and 110-pound dynamo, Ellen MacArthur from Derby, England. Ellen, skipper of Kingfisher has picked up 600 miles on Desjoyeaux and briefly was in the lead two days ago. It looks like it's turning into a drag race to the finish line between these two leaders. I'm hoping that Ellen will prevail and become the first woman to win the Vendee Globe. GO ELLEN! For more information on the race, Ellen and direct daily (Real) audio updates from Ellen herself, click:
http://www.kingfisherchallenges.com/

Design Analysis Ratios

Many analytical formulas exist which are supposed to shed light on various aspects of boat design. However, with the exception of the Prismatic Coefficient, at this time we’ll deal only with those which can be calculated and interpreted by anybody using the simple data normally included with the publication of a particular design. Moreover, I’m going to ignore those that yield limited information, one of which is the Ballast/Displacement ratio. The B/D is supposed to indicate stiffness of a particular hull, but it really doesn’t. All it shows is whether the ballast related to the overall displacement of the boat is or is not within the average design envelope-or not.

Terms used in the formulas

fig01.gif (4342 bytes)

Speed to Length Ratio

For some reason, the first ratio we usually encounter is the S/L, which is written:

fig02.gif (1848 bytes)

This is the theoretical hull speed for a displacement hull. Most sailboats are classified as displacement hulls. It is a function of the length of the wave created by the boat as it moves through the water. Wave speed is a function of wavelength. The longer wavelength is faster. Longer boats make longer waves. Since longer waves are faster, boats that make longer waves are faster. The 1.34 factor used in the equation represents the maximum speed that a displacement hull can be pushed or pulled through the water BEFORE the force required to make the hull go faster becomes inordinately high. In fact, 1.34 is what some call the breakaway point where the stern of the hull drops into the hole made by the stern wave, and the bow tries to climb the bow wave and attempts to surf as a power boat does. In the case of a displacement hull, the shape and strength of the stern sections of the hull largely prevent surfing under normal conditions, however racing boats with strong aft sections often do surf, but the forces applied to the hull are also relatively massive.

The Prismatic Coefficient

The next Design Analysis Formula usually encountered is the so-called Prismatic Coefficient. I say "so-called" because I’ve always felt that the name of this formula is confusing and should be renamed the Prismatic Percentage. I include the Cp (symbol for the Prismatic Coefficient) in this discussion because while the information published with a design usually is not enough to calculate this number, the Cp is often included in the list of specifications.

While not particularly important knowledge for the casual reader, the Prismatic Coefficient is the most important formula for the sailboat designer. The Cp compares the volume of the hull to that of a prism having the same shape for the full length of the waterline as the largest actual section. The coefficient reflects the "percentage" of the original prism that remains after the hull is carved out. The formula for the Cp is:

fig03.gif (1652 bytes)

Perhaps the following drawing will be of help in visualizing:

fig04.gif (6626 bytes)

In other words, the number generated and called the Cp show the relative "fineness" or "fullness" of the ends of the hull under consideration. In sailboats, fineness or fullness indicates a boat that is either sleek and weatherly or tubby and not efficiently close winded. Sailboat hulls with a fine entry go to weather better, but are slower at the top end than hulls that are fuller. In fact, the resultant prismatic index bears closely on the potential speed to length ratio (see above) of a specific hull. For example, all things being equal, a hull with a Cp of .52 would have a S/L ratio of 1.0. A hull with a 36 ft. waterline would have a maximum theoretical speed of 6 Knots. However, a hull with a 36 ft. LWL and a Cp of .62 would have a maximum theoretical speed of 7.8 knots, about a 20% increase. However there are other factors to be considered, and depending on what the designer desires from a design, a tubbier hull that is potentially faster may not be what is required for the job at hand. Those considerations are really up to the designer, though, and that’s why the Prismatic coefficient isn’t of much help to the casual reader. Not only that, but the person reading the design data also must know (like have a chart in his head) the Speed/Length ratio relationship as applied to the Cp numbers in order to interpret the number relative to a given design.

For the designer, however, by transposing the Cp formula, the designer can mathematically design a hull to very precise perimeters-or in the case of modern hull line generation programs, can vary the prismatic and simultaneously physically visualize (on the screen) the effect of changed hull shape and speed potential.

Displacement to Length Ratio (D/L)

Fundamentally, the D/L formula produces a non-linear index number ranging from about 40 for Ultra Light Displacement Vessels to over 500 for the very heaviest boats. What the ratio is supposed to indicate is how heavy a boat is relative to its waterline length, and by interpretation how comfortable (or uncomfortable) it might be in a seaway, and how a particular boat with a particular number may compare with a similar boat with similar index. The inference is if you have experience with a boat having a D/L of 250, then you should know about what any boat with a similar D/L might feel and perform like, but this is only true if the length of the LWL’s of both hulls are the same. The flaw in this formula, confusing particularly for newcomers to the idea is that two boats completely different in length can have comparable D/L indices, but will feel and perform at opposite ends of the spectrum. For example, take a 20 ft. keel boat with a D/L of 50, and you’ll have a boat that’ll jump all over the place and be totally uncomfortable and unacceptable in any body of water other than a mill pond. On the other hand, you can have a 60 ft. boat with a D/L of 50 and the motion and pitching moment not only may be very acceptable even in rough weather, but overall quite comfortable in general. A good example of this is the boats currently used in the Vendee Globe. I don’t have any stats for Vendee boats (Finot and the other designers are pretty secretive about those numbers), but I’m sure that based on what I see and what the skippers complain about in terms of comfort, I’d say that the D/L’s are well below 100, possibly even less than 50.

So, in order to be useful to the individual, he/she needs to have a reasonable amount of hands on experience (the more the better) with many different sailboats. Then by relating the D/L’s of those boats to the LWL and D/L number of new designs you can get a good idea of what that new design will be "like".

Regardless of these discrepancies and inconsistencies, though, the D/L is a good index albeit non-linear. The best way to get am understanding of how it all relates is to run the formula for every boat you’ve ever sailed and plot it both on a spread sheet and on an X-Y graph/chart then when you see a D/L index number for a new hull and know what the LWL is, your mind, in concert with your charts and past experience, will fill in the blanks and you’ll have a "knowing" about that new boat under scrutiny. The D/L Formula is:

fig05.gif (1550 bytes)

Sail Area Displacement Ratio
        (Power to Weight Ratio) SA/D

This formula simply supplies a (linear) number between 8.0 and 25.0, which expresses the amount of power of a boat’s working sail plan and relates it to the displacement of the boat. The SA/D formula is:

fig06.gif (1373 bytes)

The SA/D is generally thought of in context with the D/L. Normal cruising sailboats have a SA/D index ranging between 15 and 16. In general, the higher the index becomes, the more disproportionate the sail plan looks in relation to the hull. Too large a sail area can make the overall design look grotesque and out of proportion, and be hard to handle in operation. Like everything else, there are exceptions to the rule and even as applied to the visual look of a boat.

For example, America’s cup boats often have a SA/D of 25+. The reason 12 meter boats get away with these huge rigs is because of the narrow range of conditions these boats race in. In the last Cup in the Harukki Gulf in New Zealand, race conditions ranged from flat calm to about 20 knots. On race days when winds started pushing speeds of 20 to 25 knots, the races were postponed or cancelled by mutual agreement for no reason other than to protect the boats from harm (not nice to break a mast costing a million dollars-the Italian boat Prada did lose one, and America First almost did). America’s cup boats do not have reefing gear-what blows is what you gotta sail in.

Anyway, getting back to earth, a 35 ft. LOA cruising sloop designed for use by us normal mortals might have the following ratios. D/L: 270, SA/D: 15.7, Cp: .60, S/L: 7.1 kts.

Based on that what might you have? Try a conservative coastal and offshore cruiser with moderate overhangs designed in the late 60’s by Sparkman and Stevens capable of good performance, fairly fast (as boats that size go), reasonable comfort in a seaway and acceptable motion even in heavy weather. In other words, you’d have a boat that resembled something designed with the old CCA rule in mind.

In reading back over this, I see the column has gotten longer that it should. Drat! I wanted to discuss Ted Brewer’s Motion/Comfort Ratio, the Water Plane Loading Ratio, and the Capsize Screening formula, but those will have to wait for a future column. In the meantime, now that you understand all the above, and how to interpret the calculations and INTERPOLATE the numbers (weighted by personal experience) between the various ratios, especially the D/L and SA/D, you can break out your scientific calculators and try out some number crunching. OR, you can click on the following link, which will take you to "Carl’s Sailing Calculator". Plug in the numbers for any design you want to find out about and all the answers will be done for you automatically! Below is a screen shot of the calculations for TestBench, the design example I used in last month’s column.

http://www.image-ination.com/sailcalc.html

  tbstats.jpg (95364 bytes)

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