M.B. Marsh Marine Design - costs
http://marine.marsh-design.com/category/boats/costs
enSailboat rigging costs scale disproportionately with size
http://marine.marsh-design.com/content/sailboat-rigging-costs-scale-disproportionately-size
<div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even" property="content:encoded"><p>Size can be deceptive, particularly where a boat's purchase and operating costs are concerned.</p>
<p> </p>
<p>Let's consider the cost of a sailboat's rig. We'll assume that the annual cost of maintaining and repairing the rig is a fixed fraction of the cost of building the rig in the first place, and this fraction does not change with size- an assumption, yes, but likely a reasonable one for yachts of typical design and modest size.</p>
<p> </p>
<p>TL;DR: The rate at which you spend money on a sailboat's rig increases faster than the increase in boat size. This expenditure can be minimized by designing long, slender, efficient hulls that can be driven to good average speeds with relatively small rigs compared to shorter, fatter boats of the same size.</p>
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<p><!--break--></p><p>I'll use a bit of basic math in this article to illustrate where these conclusions come from. If you're terrified of exponents, just skip to the conclusions. If you see TeX code instead, you're probably blocking the MathJax Javascript library which loads from cloudfront.net. And, in case any engineering geeks are reading this: I'm talking about approximations, scaling laws and general trends. This is NOT a complete treatment of the subject, and you can't actually scale a real design in this greatly simplified way.</p>
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<h3>Scaling Laws</h3>
<p> </p>
<p>The vast majority of sailboats out there are sold by length- usually overall length, although there are <a href="http://marine.marsh-design.com/content/length">several common ways to define a boat's length</a>. Arguably the truest measure of a boat's size, though, is its<a href="http://marine.marsh-design.com/content/displacement"> displacement</a>- the actual mass of the water it displaces, and therefore the mass of the boat and everything in it.</p>
<p> </p>
<p>Consider, for a moment, how things scale with length ($L$) for boats of essentially similar design, if we change beam and draught proportionately with length:</p>
<p> </p>
<ul>
<li>Speed scales as $\sqrt{L}$</li>
<p> </p>
<li>Sail area scales as $L^2$</li>
<p> </p>
<li>Displacement scales as $L^3$</li>
<p> </p>
<li>Righting moment scales as $L^4$</li>
<p>
</p></ul>
<p>Or, thinking in terms of displacement ($\nabla$):</p>
<p> </p>
<ul>
<li>Speed scaes as $\sqrt[6]{\nabla}$</li>
<p> </p>
<li>Length scales as $\sqrt[3]{\nabla}$</li>
<p> </p>
<li>Sail area scales as $\nabla^\frac{2}{3}$</li>
<p> </p>
<li>Righting moment scales as $\nabla^\frac{4}{3}$</li>
<p>
</p></ul>
<h3>Mast & Standing Rig</h3>
<p> </p>
<p>The loads for which a sailboat's mast and standing rigging are designed come from the maximum righting moment of the boat, which scales as $\nabla^\frac{4}{3}$.</p>
<p> </p>
<p>The length of the mast and of the standing rigging scales roughly as $L$, or as $\sqrt[3]{\nabla}$.</p>
<p> </p>
<p>The major design criterion for a conventional stayed mast is whether it will <a href="https://en.wikipedia.org/wiki/Buckling">buckle in compression</a>. To a first approximation, we can look at Euler's formula for buckling, which tells us that a structural column made of a material with modulus $E$ and whose cross-sectional shape has an area moment of inertia $I$ will fail under a force $F=\frac{\pi^2 E I}{(K L)^2}$. ($K$ is a constant indicating how the top and bottom ends are secured.) If you work the exponents, you'll find that the moment of inertia $I$ scales approximately as $\nabla^2$, and the cross-sectional area of the mast scales linearly with $\nabla$.</p>
<p> </p>
<p>The cost of our mast, then, scales as $\nabla \times \sqrt[3]{\nabla} = \nabla^\frac{4}{3}$. Or, in practice, somewhat more, as the economic laws of supply and demand dictate that the smaller, more popular and easier to make mast profiles will be considerably cheaper per kilogram than larger ones for which there are fewer suppliers able to do the job.</p>
<p> </p>
<p>The cost of standing rigging can reasonably be expected to scale with the cost of the mast. Overall, it looks like we can expect the cost of the mast and standing rig to increase somewhat more than the increases in displacement as we look at larger and larger versions of a particular boat. We'll also see noticeable jumps in price when we have to add a second pair of spreaders and shrouds, and eventually a third pair, to prevent the mast from getting so thick that it completely destroys the air flow around the sail.</p>
<p> </p>
<h3>Sails & Running Rig</h3>
<p> </p>
<p>Assuming we want to look at boats in approximately the same performance class, the sail area to displacement ratio $\frac{SA}{\nabla^\frac{2}{3}}$ will be approximately constant. The sail area, then, can scale as $\nabla^\frac{2}{3}$.</p>
<p> </p>
<p>The larger boat will be faster (we saw earlier that speed scales roughly with $\nabla^\frac{1}{6}$) and the pressure of the wind on a fixed area of sail increases with the square of the speed. Let's make a (huge) assumption that the wind speed on the sails will scale roughly with the boat speed, and find that we have a $\nabla^\frac{1}{3}$ scaling of wind pressure on a sail whose area is scaling as $\nabla^\frac{2}{3}$</p>
<p> </p>
<p>In other words, the force on the sails (and therefore on the running rigging) is, approximately, a linear function of displacement: boats which perform similarly tend to have similar power-to-weight ratios.</p>
<p> </p>
<p>What this means for the cost of the sails will depend very much on the particular design. Sail area increases at a slower rate than displacement as we move to larger boats, but the stresses on those sails increase linearly- and increased stresses call for more, and more exotic, materials and construction techniques. I doubt you'd be able to keep the increase in sail and running rig cost below a linear scaling with displacement; in all probability, it'll be linear or higher.</p>
<p> </p>
<h3>Hardware</h3>
<p> </p>
<p>The loads on winches, blocks and other sail handling hardware are dictated by the forces on the sails, which as we saw above are roughly a linear function of displacement.</p>
<p> </p>
<p>A quick regression analysis on my <a href="http://www.marineoutfitters.ca/index.cfm?category=10007|10210">favourite local chandlery's price list </a>confirms that the cost of winches, blocks and such is pretty close to being a linear function of the hardware's rated load. So, at a first glance, you might think that hardware cost will be a linear function of displacement.</p>
<p> </p>
<p>Such a conclusion ignores the fact that a bigger rig doesn't only need <em>bigger</em> hardware, it also needs <em>more</em> hardware. The hardware cost vs. displacement curve is linear over small regions, but has all sorts of jumps where you have to move from single- to two- or three-speed winches, where you have to add additional hardware to handle things that would be done by hand on a smaller boat, and eventually where you have to add electric motors or 'coffee grinders' to enable a small crew to manage all that rigging.</p>
<p> </p>
<h3>And the bill comes to...</h3>
<p> </p>
<p>As a very rough approximation, then, it is reasonable to conclude that the cost of rigging a sailboat will never scale lower than the change in displacement, and is very likely to increase at considerably above the change in displacement- probably on the order of $\nabla^\frac{4}{3}$. A boat that's twice as big will cost considerably more than twice as much to rig.</p>
<p> </p>
<h3>Can we beat the math?</h3>
<p> </p>
<p>Yes, we can.</p>
<p> </p>
<p>Let's pick the size of boat we need to fit us and our stuff on the kind of cruise we want to do- 9 tonnes of fin keel monohull, for example- and consider a "typical" example of the class.</p>
<p> </p>
<p>Now stretch out the boat's length a bit, keeping the displacement constant. This will make her narrower and reduce her righting moment.</p>
<p> </p>
<p>Her speed potential will increase, due to the reduced wave drag of the longer, slimmer hull. It's easier to design a clean, low-resistance hull if it's long and slender than if it's wide and stubby, which will reduce the power needed to achieve a given speed.</p>
<p> </p>
<p>Reduced righting moment means that the longer, slimmer boat can't hoist as much sail area, and so she gets a smaller, less costly rig than her wide-beamed sister. The reduced resistance of the longer and more efficient hull, though, allows her to maintain the same average speed despite flying less canvas.</p>
<p> </p>
<p>A smaller rig means lower loads on the deck hardware, so we can use less costly winches, travellers and vangs to manage it. It'll be easier on the crew, and pushes back the margin before we'd have to pay for power assist.</p>
<p> </p>
<p>But it's still the same size of boat; we haven't changed the displacement. The livable volume is unchanged, the systems requirements are unchanged, the cargo-carrying capacity is unchanged. The cost of the hull is unchanged; after all, there's the same amount of boat there. All we've done is to reshape it into a more efficient package.</p>
<p> </p>
<p>I'd love to cite <a href="http://setsail.com/history/">Dashew's Sundeer and Deerfoot line</a>s as examples, but as luxurious premium-priced models, they don't illustrate the cost paradigm I'm talking about here- even though their performance unequivocally demonstrates the advantages of the long, slender, high-efficiency approach. Perhaps the best example- one derived directly from the mathematics I've just outlined- is the <a href="https://www.morganscloud.com/category/boat-design-selection/bds-indepth-adventure-40/">Adventure 40 that's currently taking shape over at Attainable Adventure Cruising</a>.</p>
<p> </p>
<h3>What's the downside?</h3>
<p> </p>
<p>Packing as much boat as possible into a given length means that length-based fees, such as the slip rent charged by marinas, is minimized. If we stretch out a boat's length, we see a linear increase in its slip rent. If that's your limiting factor, you're better off with the shortest, fattest tub you can find. If you don't pay slip rent by length, though, and are more interested in a boat that's economical to sail, a long, slim, efficient design may be a better bet for you.</p>
<p> </p>
</div></div></div><section class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">Topic: </h2><ul class="field-items"><li class="field-item even"><a href="/taxonomy/term/1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Boats</a></li><li class="field-item odd"><a href="/taxonomy/term/2" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Boat Design</a></li></ul></section><section class="field field-name-taxonomy-vocabulary-2 field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">Boats: </h2><ul class="field-items"><li class="field-item even"><a href="/category/boats/hardware" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">hardware</a></li><li class="field-item odd"><a href="/category/boats/sail" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">sail</a></li><li class="field-item even"><a href="/category/boats/rigging" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">rigging</a></li><li class="field-item odd"><a href="/category/boats/costs" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">costs</a></li></ul></section>Wed, 05 Dec 2012 13:53:22 +0000Matthew231 at http://marine.marsh-design.comhttp://marine.marsh-design.com/content/sailboat-rigging-costs-scale-disproportionately-size#comments