center of flotation calculation and implications?

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

  1. terhohalme
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    terhohalme BEng Boat Technology

    Dellenbaugh angle is now as accurate as ever. Computers can't change a bit of it. :)
  2. Eric Sponberg
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    Eric Sponberg Senior Member

    Bruce Number

    This week we venture into the world of sailing multihulls with a discussion of Bruce Number, which is basically a Sail Area/Displacement ratio. Unfortunately, you don’t find this in any of the classical design texts like Principles of Yacht Design or Skene’s Elements of Yacht Design, because those books don’t cover multihulls. But Bruce Number is quite important to multihull designers and enthusiasts.

    First of all, who was Bruce? Well, you maybe have heard of the Bruce Anchor or perhaps the Bruce Foil? Same guy. Edmond Bruce was British, an active member and prolific writer for the Amateur Yacht Research Society (AYRS) in the United Kingdom which published his seminal book, Design for Fast Sailing, publication #82, 1976, written with co-author Henry A. Morss, Jr. In this book is described the Bruce Number:

    BN = SA^0.5/Displ^0.333

    SA = Sail Area in square feet
    Displ = displacement in pounds

    In words, the Bruce Number is the square root of the sail area divided by the cube root of the displacement. It is not dimensionless, and the units are imperial for a reason, feet per pound. Bruce felt that a Bruce Number should approximate the boat’s speed ratio compared to the true wind speed. That is, a Bruce Number greater than 1.0 meant that the boat speed could exceed the true wind speed on some points of sail in some conditions. Obviously, this thinking will not hold with metric units. However, if using a metric version of Bruce Number consistently for comparison of different designs, then it can be valid merely to compare the power-to-weight ratios of different designs. That’s all that Bruce Number is—a power-to-weight ratio.

    I have to admit to being not as completely versed in multihull design as perhaps some of you multihull enthusiasts are, and you may know a lot more about Bruce’s work than I do. Those who are more interested in reading about Bruce’s multihull research can purchase his book from the AYRS. Another thread on this forum has useful multihull performance information, at this link:

    That is, about the first half of the thread is interesting; the second half descends into bickering among the posters, so you can ignore that.

    Another interesting and informative website is Multihull Dynamics Inc., whose link is here:

    Bruce Number has a few significant drawbacks that prevent it from being more comprehensive than you may think. In addition to sail area and weight, a sailing multihull derives power from the distance between the hulls (wider hull-to-hull beam = more stability = more power) and from the length-to-beam ratio (higher length-to-beam ratio = less wave making drag = more power). But these factors do not appear in the Bruce Number equation. So if you are comparing two multihulls of the same length and weight so that they have the same Bruce Number, the boat with the wider spread between the hulls will have more power and potentially faster speed. Likewise, a boat with narrower hulls compared to a sister generally will have less hull drag and, therefore, more speed. Therefore, Bruce Number has to be used judiciously in order to make valid comparisons.

    I used Bruce Number recently in a design project on my desk right now. A client wants to build his own 24’ catamaran, and he wants to go really fast. He does not want to race; he just wants the thrill of fast speed. He also wants to build the boat himself so that he can hone his boatbuilding skills. This is a stepping stone to a grander project. The next boat will be a 35’er that he can use for charter sailing and fishing. Finally, he wants to build a 70’er for a charter fishing business in the Caribbean. This process will take some years, but the client is only in his late twenties or early thirties, so he has plenty of time.

    Unfortunately, in my research of comparable designs, there just weren’t that many at that size to compare to. Most beachable multihull designs range from 12’ to 20’, and between 23’ and 30’, I found four Stiletto models and two RC models. So I did a parametric analysis on the boat listings that I could find and made various plots. See the figures in the attached pdf files. These are:

    Sail Area vs. Length
    Displacement vs. Length
    Sail Area vs. Displacement
    Sail Area/Displacement Ratio vs. Bruce Number

    I also attach the spreadsheet so that you can see the original data and charts for yourselves.

    These data in the spreadsheet are all published figures, and I did not try to verify any of the specifications independently. My goal was simply to find out that if I want to design a 24’ catamaran that can go really fast, what weight and sail area ball park should I be in? From the first three plots, the scatter of the data is all over the place, although the first plot of Sail Area versus Length is the tightest. But once you address weight, consistency goes out the window. However, I know I am going to be in the realm of the Stiletto and RC catamarans, and the other designs at the lower end of the scale help to give me some context as to where the design should be. Interestingly, the Stiletto boats, which are all made using carbon fiber pre-prepregs and Nomex honeycomb—meaning really lightweight construction—are in fact really quite heavy boats and not so lightweight at all.

    The final plot is actually almost an identity—two factors almost exactly the same plotted against each other. These are Sail Area/Displacement ratio, which we talked about before, versus Bruce Number. One would expect a very tight plot like this. In all these plots, I have labeled the exemplar boats (Stiletto and RC) so that I can place my new design in context with them. If boats of the Stiletto and RC weights are possible, then I am confident that my client can build his boat at my intended weight with conventional composite materials.

    So, what do I draw from these plots? I decided that I did not want to go too far out on a limb because my client is a novice, and so I chose a Bruce Number of 2.5. This puts me squarely between the RC 30 and the RC 27 by comparison. I know my client is going to be experimenting with this project. If the first boat is not right, he is ready to modify it or build a second or a third one. The big variations that he is equipped to toy with are beam spacing between the hulls and sail area. So Bruce Number leads me to a starting point where I know he can build a reasonably lightweight boat that can go fast. I attach a rendering of the hull that I finally developed. Picture two of these hulls side-by-side in a spread that has yet to be determined.

    At one point, my client and I talked about chined hulls and lifting strakes, and I recommended we wait on those concepts until he has been around the block at least once with building a basic design. Once we get some time on the water with what he can build, then we can do systematic changes to improve performance.

    In this instance, Bruce Number proved useful because it gave me a snapshot of the current market in small beach-type and trailerable cats. This is a tool to evaluate initial performance so that I can place my design in an area where top performance is possible, without being too dangerous.

    And so ends the discussion for today. I welcome any input from other multihull enthusiasts who care to shed some more light on this topic.

    We are approaching the end of this discussion thread on the design ratios. I have two topics left, and they involve speed and power calculations for powerboats. These are the Displacement Speed Formula and Crouch’s Planing Speed Formula which I will cover in the next (and last) two weeks. I have found these formulas to be particularly useful in my work, and so would like to review them here.

    When this series is over, I will collect all the lectures (12 of them) in a master pdf document and post it here so that you can download it as a complete collection.



    Attached Files:

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  3. Eric Sponberg
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    Eric Sponberg Senior Member

    Displacement Speed Formula

    Taking another 90º turn in naval architecture (from sailing monohulls two weeks ago, to multihulls last week, and now to powerboats and motoryachts), we’ll cover the displacement speed formula. You might well ask, “Which displacement speed formula—there are several of them.” Right you are, and the one I am referring to comes from Dave Gerr (pronounced like "bear" as in black bear or grizzly bear, not “Geer,” as in transmission gears). Dave Gerr, as you may know, is a very well known boat designer and the director of the Westlawn Institute of Marine Technology, the correspondence school in boat design.

    A second question you might ask is why are we breaking away from design coefficients and considering speed and power? Good question. First of all, we have just about used up all the good design coefficients, and secondly, as boat designers and naval architects, we have to be aware of the relationships between boat speed and the power required to reach a given speed—that’s the whole point of designing, building, and using boats: to travel over the water at some speed. One very well known naval architect once said that “Performance is boat design objective number one.” He was right—no matter what else your boat design is supposed to be or to do, it has to move well through the water. Think about it—if a boat does not move well, it’s useless, and there isn’t any other design feature that will ever compensate for poor performance, really. Note that I do not say that the boat has to be the fastest or to move efficiently, but only that it has to move well. That is, the boat has to achieve a speed that meets its design objective.

    Also, it is my belief that any naval architect or boat designer worth his salt could conceivably design any sort of craft because he understands the physical, mechanical, and engineering principles involved. He may not choose to design certain craft, but he could if he wanted to. And ladies, I use the pronoun “he” generically—what I say applies to you as well; you are included. A good naval architect or boat designer will be well versed in performance criteria, whether they apply to powerboats or sailboats.

    Unfortunately, the following discussion is not covered very well in either Larsson and Eliasson’s Principles of Yacht Design or in Kinney’s version of Skene’s Elements of Yacht Design. But fortunately, Dave Gerr published his displacement speed formula in his excellent book, The Propeller Handbook, by International Marine in 1989. By the way, I’m the guy that wrote the WoodenBoat magazine review of this book, issue #92, wherein I said, “Buy two copies, because the first one will probably wear out fast.” I meant it—if you don’t have this book, go buy it. Now.

    In The Propeller Handbook, Chapter 2, “Estimating Speed,” on page 10, Gerr states his displacement speed formula:

    SL ratio = 10.665/(LB/SHP)^0.333

    (and let’s call this formula “Version A”, for reasons that will become apparent shortly.)


    SL = Speed/Length Ratio = V/Lwl^0.5 (which we covered earlier)
    LB = Displacement in pounds (and let’s use our earlier designation “Displ”)
    SHP = Shaft Horsepower at the propeller
    V = Boat speed in knots
    Lwl = Length on the waterline in feet

    Gerr qualifies this equation by saying it is useful for predicting displacement and semi-displacement speeds, and he provides a chart of SL Ratio versus LB/HP with delineations of displacement speeds (up to SL Ratio = 1.4) and semi-displacement speeds (up to SL Ratio = 2.9). He further says that it is assumed that the propeller has an efficiency between 50 and 60 percent, “with 55 percent being a good average.” Yes, that is pretty average—almost all conventional propellers center around 55% efficiency.

    We know that speed/length ratio is not dimensionless, and in order for this equation to work, the other side of the equal sign must be of the same units as speed/length ratio. That’s where the coefficient 10.665 comes in. Part of the coefficient’s role is to hold all the conversion factors to make both sides of the equation equal each other in consistent units. But the other thing that is important to this equation is that 10.665 depends a little bit on what kind of boat you are looking at. For example, this value may be the appropriate numerator for lobster-style powerboats, but not so for twin-screw motoryachts, or vice versa. You have to be careful. When comparing two different boats, particularly of different sizes, they should be of the same family or hull style. Use the equation with caution.

    For example, when I was designing the Moloka’i Strait motoryachts, which are very heavy displacement, we made very good speed predictions for the MS 65 with the model testing that we did at the Institute for Marine Dynamics in St. John’s, Newfoundland, Canada, under the guidance of Oceanic Consulting Corporation. During later variations in the Moloka’i Strait designs at other lengths, from 58’ to 85’, I was able to make quick predictions of required power using the displacement speed formula rather than go back through more complicated test data and calculations. But to be careful, I recalculated the coefficient.

    On the MS 65, displacement was 181,000 lbs, SHP (actually rated BHP) was 440 HP, and Lwl was 56.58’. It is OK to use BHP instead of SHP, so long as you keep your factors consistent, and you have to assume that the drive trains are going to be similar—similar transmissions, shafting, and bearings. I knew that with these values we had indeed achieved hull speed, SL ratio = 1.34. But when I calculated the other side of the equation, the speed length ratio came out greater:

    10.665/(Displ/BHP)^0.333 = 10.665/(181,000/440)^0.333 = 1.434

    Which was too high—both sides of the equation did not equal each other. So I recalculated the coefficient:

    Coeff = 1.34 x (Displ/BHP)^0.333 = 9.966

    Part of this change in coefficient accounts for the fact that I am using BHP instead of SHP, and another part is due to the fact that it was my unique hullform. However, using the new coefficient, I could solve for the BHP of the new MS 58 motoryacht which was going to be of similar hull design as her larger sister with displacement of 115,745 lbs and Lwl = 48.75’. Using my “Moloka’i Strait” coefficient of 9.996, I can reliably calculate the BHP that I need in the new design:

    SL ratio = 1.34 = 9.966/(Displ/BHP)^0.333

    Or, so that we have only one division sign in the equation, we can flip the fraction in parentheses upside down:

    SL ratio = 1.34 = 9.966(BHP/Displ)^0.333

    Solve the equation for BHP, the only unknown:

    BHP = 1.34^3 x Displ/9.966^3 = 2.406 x 115,745 / 989.835 = 281 HP

    So, I knew to start looking for engines in the 280-300 HP range, and I was confident that I had a very reliable result.

    And the speed was going to be, at hull speed:

    V = 1.34 x Lwl^0.5 = 1.34 x (48.75)^0.5 = 9.36 knots

    Well, this is all very well and good, but then two years ago, Dave Gerr changed his formula. In the June 2008 issue of The Masthead, the design newsletter from Westlawn, Dave published this version:

    SL ratio = 2.3 – (((Displ/SHP)^0.333)/8.11)

    (and so we’ll call this “Version B”)

    And, solving for speed:

    V = Lwl^0.5 x [ 2.3 – (((Displ/SHP)^0.333)/8.11) ]

    The variables are the same as before, but the equation is totally different. Dave Gerr says that this version is more accurate at SL ratios below 2.0, and it has the benefit of not requiring a coefficient that can change with hull type.

    You can access The Masthead newsletter at this link for a complete discussion of Dave’s views on displacement speed formula:

    In fact, The Masthead newsletters are open to the public domain and you can browse through and download any of the issues that you wish. Explore the Westlawn website; this is quite a good design resource.

    I have not worked with this version of the displacement speed formula yet, as I found Version A fairly easy to deal with and am used to it. You will see that in both versions of the formula, speed is dependent on the square root of vessel length, and the cube root of the quotient of vessel displacement and horsepower. Why is that? It all comes down to the definitions of horsepower and hydrodynamic force.

    The horsepower needed to drive a vessel through the water is called “Effective Horsepower”, EHP. Power is force driven at a speed, F x V: pounds times feet/second in imperial units. One horsepower is 550 lbs-ft/second. The equation for EHP for vessels is:

    EHP = (Rt x V)/325.6


    Rt = Total resistance of the vessel moving through the water, a force, in pounds
    V = Vessel speed, in knots
    325.6 is the unit of horsepower converted so that we can use knots of speed, which we are familiar with, instead of feet per second, which we do not typically use at this level of design.

    325.6 = (550 ft-lbs/sec)/(1.689 ft/sec/knot)

    We also know that any hydrodynamic (or aerodynamic) force, such as vessel resistance, can always be expressed as follows:

    Force = (CpAV^2)/2


    C is a coefficient, be it lift or drag or whatever
    p is the mass density of the fluid involved, be it air or water
    A is generally area, but it can be an function of length squared (which is area, by definition)
    V is the speed of the thing through the fluid.

    You will recall that all aerodynamic and hydrodynamic forces can relate back to this form. We don’t care at the moment what the constants or coefficients are; we only care that Force is proportional to speed squared. Vessel resistance is a force—it has the same units, pounds—and so it is proportional to speed squared. We also know from our discussion of displacement-length ratio that resistance is directly proportional to displacement. This is to be expected; they have the same units—pounds. So, we can substitute these facts into the EHP equation above, and while we are at it, let’s ignore the constants and make the equation a proportional relationship:

    EHP ~ Rt x V

    Rt ~ Displ x V^2

    EHP ~ Displ x V^2 x V = Displ x V^3

    That is, EHP is proportional to displacement times the cube of the speed.

    Solve for speed and we have to take a cube root:

    V ~ (EHP/Displ)^0.333

    But, the term (EHP/Displ)^0.333 occurs in the denominator of Version A of the displacement speed formula, and that is why EHP and Displ are reversed:

    V ~ 1/(Displ/EHP)^0.333

    Which is saying the same thing.

    Corresponding speeds here are very important, and that is why we have speed-length ratio on the left side of the equation. That is the “corresponding speed” as we learned when discussing speed/length ratio and Froude number. For resistances of different models of vessels to be the same, their corresponding speeds—speed/length ratios—have to be the same.

    In Version B of the displacement speed formula, Dave Gerr brings the term (Displ/SHP)^0.333 into the numerator. Technically, he probably should have inverted the fraction, but he goes on to change all the other numbers and the form of the equation so that he can get away from having to use a varying coefficient due to changes in hull form. Be that as it may, the formula is still a function of the cube root of a quotient of Displ and Horsepower.

    A further note on power. EHP is the effective horsepower necessary to drive the vessel through the water. This is exactly equal to the horsepower delivered by the propeller, that is, the power on the after side of the propeller blades. The power on the forward side of the propeller blades is the Shaft Horsepower at the prop, and the quotient of EHP to SHP is the overall propeller efficiency.

    ηp = EHP/SHP

    or written another way:

    EHP = ηp x SHP

    So EHP is directly proportional to SHP, and SHP, of course, is a direct function of break horsepower, BHP. SHP is the rated horsepower of the engine less the losses due to the transmission and the shafting bearings. Since these horsepowers are all directly related, we can interchange them into the displacement speed formulas and compare different vessels so long as we keep all the types of horsepowers consistent. That is, if we have a known SHP from one vessel, and we are trying to use displacement speed formulas to find the powers or speeds for a different vessel, we will be using or finding SHP for the new vessel. Similarly, if we start with BHP of the known vessel, we will be using or finding BHP for the new vessel.

    You are encouraged to read more on horsepower and speeds. Review The Masthead newsletter in more detail. Pay particular attention to the fact that speed-length ratio—corresponding speeds—is very important, and that the displacement speed formulas apply to vessels only going at displacement speeds, and maybe at a stretch, semi-displacement speeds.

    You will also see in The Masthead a discussion of Wyman’s displacement speed formula. This was first published in the August/September 1998 issue of Professional Boatbuilder magazine, issue # 54, by naval architect and professional engineer David Wyman, where you can read the complete description of his formula as well. Wyman’s formula also gets away from having to use coefficients varying with hull form, and actually, it is basically identical to what is known as Keith’s formula for power. We’ll cover that next week along with Crouch’s formula in our last session.

    That’s quite a lot of information. Questions?

    Last edited: Mar 19, 2010
  4. Landlubber
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    Landlubber Senior Member

    I am absorning all this like a dry sponge, it takes time, but certainly really appreciate your little "talks", very kind of you to do this for us as I know you must be busy.

    Thanks from all of us here that need this information explained in more simplistic terms, we are not all maths scholars unfortunately, but I am sure anyone that wishes to understand now has the information required. Ta, John
  5. LyndonJ
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    LyndonJ Senior Member


    This is a simple smooth water prediction ?
    What happens with wave resisatnce? Are there any good rules of thumb?

    I have seen resistance speed curves which are quite different to the usual exponential growth curve for resistance. And different curves for different wave energy spectra and headings. They are always massively well above the smooth water resistance curves.

    Just wondering how useful the the simple formula really is for boats that operate at sea.
    Choosing a speed of say 6 knots for a 50 foot waterline suggests a very small power plant say 35 hp but in reality it will need 80 minumum to motor to windward at 3 knots in just a force 5 , and it's the wave response not the wind that will cause this.

    And nothing for the windage of a sailboat, Gerr suggests dropping the speed by 2% but in reality it's going to be very dependent on projected areas.
    Wouldn't it be better to design not only for smooth water but also assessing the ability for motoring into a gale?

    So do you design for an SL of 1.3 and power for that and accept that wave resistance will reduce that speed to 30% when going full tilt to weather ? A lot of work boats being driven so fuel hungrily at an SL of 1.5 or so with their bows in the air actually only achieve an SL of 1 with that same power in the teeth of a moderately strong wind and a long fetch.

    I always read these sorts of guides and think "YES" a simple solution then you look at RAO's and realise smooth water prediction is not much use really unless you operate on sheltered waters.

    Very interested to know what you think.
  6. Eric Sponberg
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    Eric Sponberg Senior Member

    Thanks, Landlubber, glad you are getting good information from this. As I recall, it was you who suggested the series, so thank you for that.

    You are right, these formulas are for calm water performance. When it comes to waves, it is anybody's guess as to what power is required for speed in heavy weather. Some waves are taller and steeper (i.e. "squarer") than others. Some vessels have more top hamper than others and so they will behave differently in high winds. The angle that the vessel makes to the wind and waves (angle of encounter) plays a huge roll in heavy weather resistance and powering. You end up with having to map a whole spectra of conditions, and that usually is beyond the scope of small boat design.

    Kinney, in Skene's Elements of Yacht Design, suggests that you establish the calm water speed/length ratio that you want to go, determine the installed horsepower, and then increase that horsepower by one third to account for heavy weather conditions and bottom fouling over time. That works well.

    Larsson and Eliasson's Principles of Yacht Design covers added resistance in waves in much more detail, but they admit that added resistance changes greatly with conditions. Which conditions do you want to design for? Take your pick. Then of course, when are you going to actually see those particular conditions--maybe not often. It is worthwhile to be aware of added resistance in heavy weather, and it is quite another thing to design specifically for it. This problem is much easier to quantify with known average weather conditions for vessels that serve a particular route, like commercial ferries or offshore supply vessels.

    One thing I noted in the Moloka'i Strait vessels, when we were considering a number of different engines, is that some mechanically controlled engines did not have too much excess power above the propeller curve, which suggests that they would not be good candidates for achieving speed in heavy weather when resistance goes up. They would not have the power to push through. But the newer electronically controlled engines have tremendous excess power over the propeller curve at low RPMs which tells me that they could handle added resistance quite well, and still keep moving at reasonable speed through heavy weather.

    You can always expect that your boat will travel more slowly in heavy weather, and that the crew should be judicious in picking its route and speed. You cannot expect to make your calm water design speed in heavy weather. Slow down, pick another angle of encounter, or simply don't go--wait for the weather to change; it always does.

    Good questions, thanks.

  7. fredschmidt
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    fredschmidt Naval Architect

    Great job!
  8. MikeJohns
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    MikeJohns Senior Member

    Eric you are producing a stirling effort here. For the educational side for others reading this I'd like to add to the wave resistance factor already raised.

    It doesn't need to be even near heavy weather either. Any waves have a significant increase in resistance. You should always power well up the reistance curve for a power boat that operates out of strictly sheltered water. Then give some thought to a more efficient installation for the prop ( larger and slower turning more blades whatever...) and a more realistic top speed (lower) try and envisage the worst wave energy spectra you want to be able to power into.

    Without considering this, it may be misleading to the designer to base a sea-going power prediction on smooth water tests without considering a lot of other design factors, for example; the angle of entrance both at the waterline and well above it.

    For interest here's a simple resistance predicition for an AC 12m sailboat which is already a low resistance hull ( from one of Van Ossanaon's papers ) and this is nowhere near heavy weather, just a normal sea surface.
    You can see that there is a trap trying to opt for a low power low speed economical installation. Such designs tend to be unable to proceed directly to windward at times when they really need to.

    Any model can be tested for wave related resistance in a wave tank facility and this is what RAO's are all about. It's not cheap but the budding designer may want to consider this if the contract requires.

    It's the shorter smaller steeper waves often encountered in coastal or lake boating that cause the most resistance to boats ( as opposed to ships).

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  9. DCockey
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    DCockey Senior Member

    BN and SA/D are the same information presented with different scales. The Bruce Number is exactly proportional to the square root of the SA/D. Or the order can be swapped to the SA/D is exactly proportional to the square of the BN.

    From Eric Sponberg's informative post about the Bruce Number (BN):
    BN = SA^0.5/Displ^0.333
    SA = Sail Area in square feet
    Displ = displacement in pounds

    From Eric's earlier post about Sail Area / Displacement Ratio (SA/D):
    SA/D = Sail Area/(volume of displacement)^0.667

    The numerical relationships between them involves the density of water since SA/D is dimensionless and BN has dimensions of feet / [cube root of lbs]

    BN = (SA/D ^ 0.5) * (1 / [Density of water] ^ 0.3333)

    SA/D = (BN ^ 2) * ([Density of water] ^ 0.6667)

    For sea water at 64 lbs / cubic foot:

    BN = (SA/D ^ 0.5) * 0.25

    SA/D = (BN ^ 2) * 16.00

    SA/D BN
    10 0.7906
    12 0.8660
    15 0.9683
    20 1.1180
    30 1.3693

    BN SA/D
    0.8 10.24
    0.9 12.96
    1.0 16.00
    1.1 19.36
    1.2 23.04
    1.3 27.04
    1.4 31.36

    Eric directly computed BN and SA/D for a variety of multihulls, plotted the results and apparently used a regression to arrive at virtually the same relationship. (The difference is presumably due to various round-off errors.)
  10. Paul Kotzebue

    Paul Kotzebue Previous Member

    For those with an interest in yachting history, the Bruce Number is proportional to a sailing yacht's rating under the Universal Rule devised by N.G. Herreshoff.

    R = 0.18*L*SA^0.5/D^0.333

    L = measured length
    SA = sail area in ft^2
    D = displacement in ft^3
  11. Brent Swain
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    Brent Swain Member

    Hats of to Eric. Well done , Many thanks.
    I can count on one hand, the designers who take the time to answer design questions on these chatlines. There are far more out there . Where are they? Do those who won't give an answer, unless there is a quick buck in it, deserve our business? I think not.
  12. magwas
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    magwas Senior Member

    I am not in a position to tell anything about naval architecture, but I feel the same about my own profession.

    And thank to those few who do answer questions. I have learned a lot from this thread.
  13. Eric Sponberg
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    Eric Sponberg Senior Member

    Crouch’s Planing Speed Formula, Keith’s Formula, And Wyman’s Formula

    In our final discussion of this series, we take up one primary performance formula, Crouch’s Planing Speed Formula, and two other ones, Keith’s Formula and Wyman’s Formula which I mentioned at the end last week on the discussion of displacement speed formula. Boats travel either at displacement speeds, or planing speeds, two very different hydrodynamic regimes. There is, of course, the semi-displacement regime which is that middle region between the two, but for the sake of discussion, we will differentiate these two basic regimes of motion.

    Larsson and Eliasson’s Principles of Yacht Design, 3rd edition, discusses high speed craft, but they don’t mention anything about Crouch’s Formula. However, Skene’s Elements of Yacht Design does, as does Gerr’s Propeller Handbook. I have used Crouch’s Formula a number of times, and it always seems a little bit different each time. First of all, who was Crouch?

    George F. Crouch was a famous American naval architect in the early part of the 20th century. An 1895 graduate of Webb Institute of Naval Architecture, he went to work in industry for nearly a decade, but returned to his alma mater to become professor of math for about ten years, then ultimately became a full professor of naval architecture and resident manager of the college. In 1923, I guess he’d had enough of academia after nearly two decades, and decided to go where the money was. I may not have the order perfectly correct, but we know that in 1924, Crouch was vice president of design for Dodge Watercar, a new boatbuilding venture and brainchild of Horace E. Dodge Jr., son of one of the founding brothers of the Dodge automobile manufacturing company. Crouch held this position for a number of years. Horace wanted to race boats, and he also wanted to manufacture an “everyman’s” boat on an assembly-line basis much like his father’s automobiles, hence the name “watercar”. This was to be in direct competition with the Chris Crafts and the Gar Woods of the time, all of them centered in the Detroit, Michigan, area.

    I guess Crouch was still able to do consulting and custom design on the side because his iconic racing boat design Baby Bootlegger became the Gold Cup racing champion in 1924, and it repeated its victory again the following year. I say iconic because Baby Bootlegger really was a different stroke in boat design with three notable features: An unusual rounded shear; a canoe stern that overhung the aft end of the planing surfaces by some few feet; and an innovative wedge-shaped rudder. Many reproductions of Baby Bootlegger have been built; you can buy plans and build one yourself; and you can buy model kits of this famous racer. Baby Bootlegger probably overshadowed everything else that Crouch designed, including sailboats later in his career. Baby Bootlegger still exists and was meticulously restored about 25-30 years ago. A wonderful article about her appeared in WoodenBoat magazine, issue #60, September/October 1984. Crouch also worked for Nevin’s Boatyard in New York for a long time, where Baby Bootlegger was built, and he died in 1959.

    So what is Crouch’s Formula, which he surely used to great effect in his power boat designs? Here it is:

    Speed, V = C/(Displ/SHP)^0.5

    V = speed in knots
    Displ = boat displacement in pounds
    SHP = shaft horsepower at the propeller
    C = a coefficient depending on boat type.

    Kinney in Skene’s Elements of Yacht Design gives a general range of the value of C as from 180 to 200. This is not too helpful, but Gerr gives a better breakdown in The Propeller Handbook, namely:

    C Type of Boat
    150 Average runabouts, cruisers, passenger vessels
    190 High-speed runabouts, very light high-speed cruisers
    210 Race boat types
    220 Three-point hydroplanes, stepped hydroplanes
    230 Racing power catamarans and sea sleds

    Personally, I don’t let these coefficients color my thinking too much. I know there is a range, but what you are supposed to do is find similar boats of the type that you are analyzing or designing, and back-calculate what the coefficient C is. That is:

    C = V x (Displ/SHP)^0.5

    Then proceed with that coefficient with the boat that you are analyzing or designing. That is, C is very much an empirical number—it changes all the time. Note that C must have units that allow the quantity 1/(Displ/SHP)^0.5 to end up in knots of speed.

    A curious thing here is that the quotient Displ/SHP is taken to the square root power, whereas last week when we talked about displacement speed formula, this same quotient was taken to the cube root power? Why the difference?

    To tell you the truth, I don’t have a good answer. I find this very curious because last week we saw that a power calculation is the product of Force times Speed:

    Power = Force x Speed, with Force in this case being the hydrodynamic drag

    HP ~ Rt x V

    Rt ~ Displ x V^2

    HP ~ Displ x V^2 x V = Displ x V^3

    That is, horsepower is proportional to displacement times the cube of the speed, and if we are to solve for speed, we have to take the cube root of HP/Displ. But Crouch’s equation, which we can rewrite thus to get rid of one division sign:

    V = C x (SHP/Displ)^0.5

    uses the square root of this quotient.

    I don’t know the answer to this quandary. Did Crouch figure that cube root was probably involved, but decided he did not want to go through the labor of calculating cube roots and settled for a square root function??? Rather than deal with cube roots to get an exact result, he maybe relied on the square root of the quotient and a sliding scale of his power coefficient, C, to make the equation work quickly and easily. Back then, he did not have nifty calculators or computers that automatically calculate numbers to any conceivable power or fractional power to the greatest degree of accuracy. But he most likely had a slide rule, and if he had a “K” scale on his slide rule, he could easily calculate cubes and cube roots. That’s why I think this is all very curious.

    Slide rules date back to the early 17th century (I still have mine from college, and still use it from time to time), and were highly refined by the time Crouch was born. (Did you hear about the constipated mathematician?? He worked it out with a slide rule. HAR, HAR—GROAN! That’s an old childhood joke.) The long and the short of it is, I don’t know the answer—a square root does not make sense in the physics of the matter. Go back through any hydrodynamic study in the last 50-60 years, and you will always find a force related to speed squared, and power related to speed cubed—there is just no way around it. So I wonder if Crouch’s formula simply was a mathematical simplification yet still have horsepower within the formula.

    So, does Crouch’s formula work? Yes, pretty well. I recall once talking to the vice president of engineering at a major muscle-boat manufacturer who claimed that he successfully brought a claim against his engine supplier for faulty engines when he discovered that he was not getting the power out of the boat that he should have been getting—his race boats weren’t as fast as they should have been as calculated using Crouch’s formula. His coefficient was supposed to be C = 225 which he had proven time and again on the race course for his style of boats. But when the test results of the latest speed runs on a new boat came back with low speeds, and knowing that the boat came in on weight and on spec so that C was still reliably 225, the only thing that had to be off was the horsepower. The engine manufacturer relented and replaced the engines, and the problem was cured.

    For myself, I have used Crouch’s formula to characterize the design at hand, or to mimic other designs when I am trying to home in on new power and weight parameters. I’ll calculate the C coefficient for known other boats of similar form, and then use that coefficient for the new design at different weight and power. That coefficient usually does not conform too well to Gerr’s classification above, and that is why I generally ignore the names—I am only interested in the value of C at hand. Once I am in the ballpark, then the design effort becomes much more specific in choosing the correct engine, reduction gear, and propeller size. For that, I rely on two sources, the first of which is the Bp-δ method (pronounced “Bee-Pee-Delta”) of propeller specifications, which is found in The Propeller Handbook and is also covered in other major professional naval architectural texts. Then I revert to more detailed calculations using the NavCad software from HydroComp Inc. in New Hampshire. More often than not, Bp-δ and NavCad come out in very close agreement. Crouch’s formula is a quick and easy way to get started with the right ballpark of horsepower and weight.

    Isn’t there a more universal formula that covers both performance regimes, you may ask? Indeed there is, two of them, in fact. The first of these is Keith’s Formula which, interestingly, is mentioned only in Kinney’s Skene’s Elements of Yacht Design, and not anywhere else that I can find. Who was Keith? I haven’t the foggiest idea. I have been researching all over the Internet and asking the most renowned powerboat design experts in the US and the world, and no one can come up with any original information or source for Keith or his formula. Keith is a total enigma. Nevertheless, here is his formula, and I mention this only because it leads us directly into the other formula that is available, Wyman’s Formula:

    Keith’s formula states:

    Speed, V = Lwl^0.5 x C x ((BHP x 1000)/Displ)^0.333

    V = speed in miles per hour (not knots)
    BHP = Break Horsepower
    Displ = Displacement in pounds
    C = Coefficient that ranges between 1.3 to 1.5

    Notice a few interesting things compared to Crouch’s formula. First of all, horsepower is Brake Horsepower (installed horsepower of the engine, before it gets to the prop), and that horsepower is multiplied by 1000. Or, you could also say that the 1,000 number is the displacement divided by 1,000. Either way, this has the effect of reducing the coefficient C down to a very low number. Note also that the quantity that includes the quotient of BHP/Displ is taken to the cube root, as we should expect.

    The other interesting thing that we see in this equation is the value of the square root of the length on the waterline on the right side of the equation. If we move it back to the left side of the equation, it goes to the denominator, of course, and that gives us speed/length ratio:

    V/Lwl^0.5 = C x ((BHP x 1000)/Displ)^0.333

    which results in exactly the form of the equation of the displacement speed formula from last week. So it is the same thing written just a slightly different way (only one division sign).

    In Keith’s Formula, Kinney does not give any good description of how the coefficient C varies, only that it does, and that you do the same thing as with Crouch’s formula—back-calculate C from exemplar boats, and use it again for more or less reliable predictions on new designs. That can be a problem if you don’t have any exemplar design to refer to, or if their weight, speed, and horsepower data are incomplete.

    To solve that, we can turn to Wyman’s Formula. Wyman’s formula is exactly the same as Keith’s Formula, it is just that Wyman has given a fair bit more effort to defining the coefficient C. First of all, who is Wyman?

    David B. Wyman is a practicing naval architect and professional engineer currently living in Maine. He trained in naval architecture at the US Merchant Marine Academy, has taught at the Maine Maritime Academy, and has done extensive work for the US Navy. He currently specializes in the design of small submersible vessels. In the August/September 1998 issue of Professional Boatbuilder magazine (PBB), issue #54, he described his formula and the derivation of his coefficient. This information is repeated in The Masthead issue of June 2008 that I cited last week, whose link I repeat here:

    Wyman’s Formula is:

    V = Cw (Lwl^0.5) [SHP/(Displ/1000)]^0.333

    V = boat speed in knots
    Cw = Wyman’s Coefficient
    Lwl = length on the waterline
    SHP = Shaft Horsepower at the prop
    Displ/1000 = Displacement in pounds divided by 1,000

    Note that it is displacement divided by 1,000, not SHP multiplied by 1,000. This is exactly the same as Keith’s formula, which Wyman admits he started with. This formula also includes the square root of length on the waterline, so it is a “corresponding speed” formula. But Wyman was disturbed that C in Keith’s Formula was not well defined, so he started to collect performance data from a wide variety of boats over the years (over 50 different designs) in order to determine an empirical value, equation, or chart for C, which he now calls Cw, the Wyman Coefficient. This he plotted against speed/length ratio, and got a surprisingly consistent straight-line relationship for Cw. A plot of his data appears in the PBB article, and a slightly different version with the axes reversed appears in The Masthead article. Cw = 0.7 at S/L = 0.0, and Cw = 2.5 at S/L = 10.4, according to the PBB plot. You can do straight-line relationships with these limits if you don’t have access to the PBB article which appears to be the more comprehensive plot of the two.

    By the way, you can buy PDF downloads of any issue of Professional Boatbuilder magazine for US$3.50. Here is the link:

    The complete series of Professional Boatbuilder magazines, always updated to the current issue, is available on a flash drive for US$145. Link:

    And just so that you know, all Professional Boatbuilder issues since #95 in June/July 2005, are on PBB’s website at: Anyone can access any issue since then, a tremendous resource.

    So, what do we do with Wyman’s Formula? The same thing that we can do with the displacement speed formula or Crouch’s Formula. Wyman’s Formula is the most comprehensive of all of them because it applies equally well to displacement boats, semi-displacement boats, and planing boats. Pick the Cw coefficient that corresponds to the speed/length ratio, and solve for the unknown that you are looking for—speed, length, weight, or horsepower. That is, Cw is always determined for you; you don’t have to go searching for it, and that is why Wyman’s Formula is so useful. It has been determined empirically from existing boats.

    Now that you see how this is done, of course, you can back-calculate the coefficient if you want to. Let’s say you have your own collection of boat designs, different from Wyman’s. You can solve the Wyman Formula for your population of boats and determine your own Wyman Coefficient. This is a way to home in a little more accurately for your own style of design. This is essentially what I have done before with both displacement speed formula, where the 10.665 coefficient was fixed but I changed it to suit my designs, and Crouch’s Formula where I again determined by own coefficient. In the end, we see that Crouch’s Formula appears to be an unnecessary simplification of what Wyman’s Formula now does with ease. So I say, let’s forget about displacement speed formula, Crouch’s Formula, and Keith’s Formula, because Wyman’s Formula does it all quite well because it is grounded in physics and is based on what actual boats do.


    Ladies and gentlemen, this is the end of the series on The Design Ratios. This has been a fun ride, and I am grateful to all of you for your compliments and comments that have been posted for the benefit of all the readers. In particular, I wish to thank Captain Vimes for his original question (Thank you, Capt.) and to Landlubber who suggested doing a whole series on design ratios (Thank you, LL).

    Within the next few days, I will post a PDF document containing all the discussions in the order they were presented, edited to clean up the text a little bit, and with the diagrams and pictures embedded within the text so that they make for a little easier reading. Some documents were posted separately, such as a few spreadsheets and an article or two, and I’ll post those again so that they are all available at the same place. Finally, I will post everything on my own website so that there is a second repository for the information.

    To all of you who have been following this series, thank you for your interest. This series means nothing if not for readers who read it. I hope that The Design Ratios has helped you to understand boat design a little better, and to all the budding designers as well as practicing professionals, I hope that the history and derivations of some of the material have been enlightening.

    Good luck to you all; it has been a pleasure to talk with you.

    Best regards,

    1 person likes this.
  14. ancient kayaker
    Joined: Aug 2006
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    Location: Alliston, Ontario, Canada

    ancient kayaker aka Terry Haines

    A magnificent effort Eric: we all thank you.

  15. TollyWally
    Joined: Mar 2005
    Posts: 774
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    Location: Fox Island

    TollyWally Senior Member

    Thank You Eric,
    I come here primarily to learn, although I often get sidetracked. This has been a wonderful series, I look forward to the pdf which will certainly be added to my reference library, both digital and paper. Thanks again.
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