M.B. Marsh Marine Design - Stability
http://marine.marsh-design.com/category/boats/stability
enDynamic Stability of a Monohull in a Beam Sea
http://marine.marsh-design.com/content/dynamic-stability-monohull-beam-sea
<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>The last post in <a href="http://marine.marsh-design.com/category/boats/stability">our series on yacht stability</a> looked at the<a href="http://marine.marsh-design.com/content/understanding-monohull-sailboat-stability-curves"> static case</a>. We saw that a yacht's response to heeling forces can be described by a stability curve, the shape of which tells us a lot about the boat's purpose, sailing characteristics and seaworthiness.</p>
<p>A real yacht in a real situation is of course far from being a static case. The static stability curve is one of our best tools for quantitatively comparing different designs, and I don't mean to denigrate its importance. In practice, though, dynamic factors can often have a bigger effect on the actual stability of the boat as it relates to safety, seaworthiness and comfort.</p>
<p><!--break--></p>
<h3>Warning: Math!</h3>
<p>Our earlier introduction to static stability was relatively light on mathematics. While some calculus is required to <em>produce</em> a stability curve, <em>reading</em> it requires only a conceptual understanding, not any actual calculations.</p>
<p>Try as I might, I can't think of any way to explain the key aspects of dynamic stability without resorting to the language of the natural world, i.e. mathematical physics.</p>
<p>If you keep reading, then, you're going to encounter equations and free-body diagrams. The actual equations of motion for a yacht in waves are horrifically complex, the sort of vicious tensor calculus that's nearly impossible to solve by hand and so is usually left to numerical approximation on a computer. (Even today, no-one has a fully satisfactory mathematical model for an arbitrary vessel in arbitrary conditions.) For the present purposes, then, we'll take a common physicist's shortcut, often described as "Assume the horse is a sphere of uniform density in a frictionless vacuum". In other words, we'll use a greatly simplified model to understand the main characteristics of the real thing, with the tacit understanding that reality will add many additional, usually smaller, effects.</p>
<h3><em>I</em>, the Moment of Inertia</h3>
<p>Stability, in the sailing context, is concerned mainly with rotary motions: the roll or heel of the boat from side to side, and the fore-aft pitching or trimming motion. The nature of these motions depends on the distribution of mass within the yacht, which we characterize with a parameter called the "moment of inertia".</p>
<p>Most folks are intuitively familiar with the moment of inertia, even if they've never heard the term. Take two pieces of wood, an eight-foot 2x2 and a two-foot 4x4. Swing each one like a baseball or cricket bat. The long thin one is harder to swing- and packs a bigger whallop when it hits something- than the short thick one. The two sticks have exactly the same mass, but the long thin one's mass is farther from the centre of rotation. It therefore has a larger moment of inertia.</p>
<p>We can state that more generally and more formally as</p>
<p>$I = m r^{2}$</p>
<p>for a point object of mass $m$ at a distance $r$ from the centre of rotation. If we consider a complex object made of many point-like components, we simply sum the moments of inertia for all the components, yielding:</p>
<p>$I = \sum\limits_{i=1}^{N} m_{i} r_{i}^2$</p>
<p>Or, extending to a large solid object whose density is $\rho(\vec{r})$ at each point $\vec{r}$, we integrate $mr^2$ over the entire volume of the object to find:</p>
<p>$I = \int\limits_V \rho(\vec{r}) \vec{r}^2 dV$</p>
<p>In other words, the moment of inertia increases if:</p>
<ul>
<li>The mass increases (twice the mass, twice the moment), or</li>
<li>The mass is placed farther from the centre of rotation (twice as far, four times the moment).</li>
</ul>
<p>From this we can see that weight in a sailboat's rig, keel and gunwales will tend to increase its moment of inertia in roll. Weight in the rig, keel, lazarette and forepeak will increase the moment of inertia in pitch. Lightening (or removing) the rig, or moving heavy gear to the middle of the cabin, will decrease the moments of inertia. We can also see that, thanks to that $r^2$ term, distance from the centre of rotation (usually around the centre of the hull and not too far from the waterline) is a hugely dominant factor- meaning that the rig, by virtue of its height, contributes a great big chunk of $I$ despite its relatively light weight.</p>
<h3>Angular Acceleration</h3>
<p>We're interested in how a boat responds to forces, and its distribution of mass is only one part of that. We don't perceive the forces on the boat directly, we perceive the accelerations they produce. To relate force, shape, mass and acceleration, we'll need two more equations. The first is the torque or moment $\vec{\tau}$, which is the twisting effect produced by a force $\vec{F}$ acting at a distance $\vec{r}$ from the centre of rotation:</p>
<p>$\vec{\tau} = \vec{r} \times \vec{F}$</p>
<p>The second relates the torque $\vec{\tau}$ to the acceleration $\vec{\alpha}$, and the relationship is- surprise, surprise- the moment of inertia.</p>
<p>$\vec{\tau} = I \vec{\alpha}$</p>
<h3>Enough equations, let's look at a boat</h3>
<p>Okay, here's a boat. She's holding a steady course on flat water with the wind on the port beam, and for now we'll consider only the two-dimensional plane in which the forces that go into a static stability curve are acting. Those forces are:</p>
<ul>
<li>$F_{sail}$, the sideways force applied to the sails by the wind. (The forward component of this force, which propels the boat, is directed into the page.)</li>
<li>$mg$, the force of gravity on the boat (equal to the boat's mass times the gravitational acceleration).</li>
<li>$F_b$, the force of buoyancy, which acts upward through the centre of buoyancy. It's offset because, as the boat heels in the wind, her centre of buoyancy shifts to leeward.</li>
<li>$F_{lat}$, the laterally-directed lift of the keel, rudder and underbody as they move through the water.</li>
</ul>
<p><img alt="" src="/sites/default/files/u4/2013/June/stability/free%20body%20static.svg" style="width: 560px; height: 560px;" /></p>
<p>In the static (equilibrium) case, these forces must sum to zero in the vertical direction and in the horizontal direction, and the net torque must also be zero. The wind pushes the sails to starboard; this is balanced by the keel and rudder pushing the hull to port. The boat's weight is balanced by her buoyancy. Finally, she'll choose her angle of heel so that the righting moment created by shifting the centre of buoyancy to leeward twists her counterclockwise in exact balance with the clockwise heeling moments that result from the sail and keel forces.</p>
<p>That's the static case, and if you work through that balance of forces for every possible angle of heel, you get the static stability curve.</p>
<h3>Top o' the Wave</h3>
<p>If you've ever been out in real waves- not just rough inshore chop, but real many-metres-high waves- you've probably felt a moment of weightlessness as the boat goes over the crest. This is not an illusion. It's the exact same weightlessness that millionaire playboys get when their Learjet pilots fly parabolic loops, and it's the exact same weightlessness that astronauts get aboard the ISS.</p>
<p>Gravity's still acting on you in all three cases, of course. But we're physicists now, so we can freely trade a force for an acceleration simply by moving to a different frame of reference. If our frame of reference is fixed relative to the Earth's surface, we perceive gravity as the usual, familiar force. If our frame of reference is accelerating towards the Earth at g = 9.81 m/s<sup>2</sup>, we perceive the force of gravity to be zero.</p>
<p>In the Learjet's case, we start by flying up, and we steer the plane in an arc that follows the trajectory a thrown object would take. The passengers inside are in free fall, being accelerated toward Earth by gravity, and we're accelerating our reference frame- the plane- in perfect sync with them. The passengers therefore feel weightless. The ISS does the exact same thing, except that it's also moving sideways so fast that the Earth's curved surface falls away from under it as quickly as it falls toward the planet. Aboard our yacht, the wave has accelerated us upward, and then- at the top of the crest- the wave starts to fall away from under us, sometimes almost as fast as gravity pulls us (and the boat) back down.</p>
<p>Here's our first big lesson in dynamic stability: Buoyancy is based on <em>weight</em>, not on <em>mass</em>. Our boat's <em>mass</em> hasn't changed, but when we go into free-fall on the crest of the wave, the boat's <em>weight</em> in her own reference frame is sharply reduced. If you suddenly feel like you weigh half as much, then the boat also weighs half as much, she's fighting back with half her usual buoyancy, and the height of her stability curve is effectively cut in half.</p>
<p>In the fully weightless case, working in the boat's own reference frame (which, as you recall, is accelerating towards Earth at 9.81 m/s<sup>2</sup>), we have this:</p>
<p><img alt="" src="/sites/default/files/u4/2013/June/stability/free%20body%20free%20fall.svg" style="width: 560px; height: 560px;" /></p>
<p>Ouch! We still have the clockwise heeling moment from the sails, and a corresponding one from the keel and rudder (now directed slightly upwards, as the water is moving up and left relative to the boat). But the buoyancy force that would fight back to keep her level is gone. The free-body diagram says she's going over.</p>
<p>Here's where we have to jump back to some math. The boat starts out with zero angular velocity. We're applying forces that will try to rotate her clockwise. They'll produce a net torque</p>
<p>$\vec{\tau} = a F_{lat} + b F_{sail}$</p>
<p>on the boat. The only thing she has to fight back with is her inertia- described in this case by the roll moment of inertia. Her rotational acceleration will be:</p>
<p>$\vec{\alpha} = \frac{\vec{\tau}}{I}$</p>
<p>So a <em>higher</em> torque or a <em>lower</em> roll moment of inertia means a greater acceleration. If we want to survive this incident, we have to keep the angular acceleration low enough that the boat finishes her free-fall, and starts fighting back with buoyancy, before she's had time to roll on her side.</p>
<h3>Changing the Yacht's Behaviour</h3>
<p>If the boat's going to be tossed around like this, we want a high roll moment of inertia. Recall that $I$ comes from the mass of each component and the square of its distance from the roll centre. That means that while the ballast and the hull do contribute, a surprisingly large fraction- often about half of the total- is from the yacht's rig.</p>
<p>A nice hefty rig, with a fair bit of weight aloft, is not good for static stability (and therefore for sail-carrying power, and therefore for performance). It does, however, add greatly to the roll moment of inertia, and therefore helps to reduce roll accelerations. That translates to a reduced tendency to be blown over when the boat is tossed over the crest of a wave (not to mention a slower roll motion at anchor).</p>
<p>We can also see that, if the boat is dismasted, the roll moment of inertia will be greatly reduced. A dismasted boat is in a risky situation indeed- not only do you have to deal with the shattered remains of the rig, you're left with a vessel whose angular accelerations will be much greater than before. Although her <em>static</em> stability is slightly improved by losing all that weight up top, her <em>dynamic</em> stability is severely compromised; without the mast, she'll be much more susceptible to wave-induced capsize.</p>
<p>This is why, in my opinion, shaving every last kilogram out of a cruising boat's rig is counterproductive. A stronger, heavier rig yields a higher roll moment of inertia, therefore a reduced roll acceleration, and is less likely to be lost over the side. The slight loss of sail-carrying power, relative to a lightweight racing rig, is barely noticeable to a cruiser.</p>
<p>Our other tool for improving this situation is to reduce the net overturning torque on the yacht.</p>
<p>Reducing $F_{sail}$ is pretty straightforward- we reef her down, and if that doesn't do the trick we can always drop the working sails in favour of a storm jib and trysail.</p>
<p>The other overturning force we're interested in is the side force on the keel. It, too, is trying to overturn the boat. We can easily imagine a situation where our boat is falling off a wave onto her beam ends, and "trips" over her keel (which is still buried in the wave). If she rolls far enough for the gunwale to dig in, the added resistance it presents makes the situation worse. We would like to avoid that, ideally by preventing the boat from going beam-on to the waves. Failing that, we'd like to reduce $F_{lat}$ as much as possible, which means retracting the underwater appendages.</p>
<h3>Skidding Sideways</h3>
<p>Now and then, someone asks "How can a centreboarder be seaworthy without a deep ballast keel?" This is a big part of the answer. If she raises her board, the centreboarder loses a big chunk of $F_{lat}$ without losing too much roll moment of inertia. In situations where a deep-keeled vessel would "trip over her keel" as she falls sideways down the wave, the centreboarder can skid sideways. Her static stability curve may appear less favourable than that of the keelboat, but in a real survival situation, being tossed around by waves, the centreboarder's improved control over the dynamic side forces compensates for her higher centre of gravity.</p>
<p>Catamaran and trimaran stability is an issue for another day, but in case you're wondering, the same effects are in play there. A cat with her boards up and sails down is very, very hard to flip; with very little $F_{lat}$ and a truly enormous roll moment of inertia, her roll accelerations are kept small and she'll tend to skid sideways down the wave. I suspect it's safe to say that most multihull capsizes are due to too much $F_{sail}$, i.e. "hot rodding" the boat with too much sail up for the conditions; a breaking wave big enough to flip a reefed-down cruising cat would be a terrifying thing indeed.</p>
<p>The preferred approach, though, is not to chance it, and avoid going beam-on to the waves. On almost every boat, the pitch moment of inertia is greater than the roll moment of inertia, the overturning forces are much less in the fore-aft direction, and the hull can cut through a wave much more cleanly bow-on or stern-on. In many cases, then, we can avoid many of the problems brought up here by using a drogue or sea anchor to keep the boat in a more favourable orientation. The same physics applies fore-and-aft, but with numbers that are more in our favour.</p>
<h3>Summary</h3>
<p>We may summarize this article in a few take-home points:</p>
<ul>
<li>The dynamic behaviour of a yacht in waves is at least as important, if not more so, as its static stability curve.</li>
<li>When we're taking heavy weather from abeam, we want to keep roll accelerations low.</li>
<li>Roll accelerations are reduced if we have a high roll moment of inertia or if we reduce the overturning forces.</li>
<li>Deep ballast and heavy rigs increase the roll moment of inertia. Dismasting greatly reduces the roll moment of inertia, and lightly rigged or dismasted yachts are generally easier to capsize than those with sturdy (and intact) cruising rigs.</li>
<li>Reefing the sails and retracting the underwater appendages will reduce the overturning forces. Anything that presents a lot of lateral resistance in the water can "trip" the boat if it gets hit beam-on by a breaking wave.</li>
<li>All of this is less of a problem if we can keep the boat bow-on to, or running with, the waves.</li>
</ul>
<p>We must keep in mind, of course, that we've been working with a grossly simplified first-order model. A lot of corrections and perturbations must be added to this model to get quantitative results for any particular boat and situation. To a first approximation, though, the conclusions drawn from the basic physics are pretty much the same as those drawn from decades of study by hundreds of designers and skippers.</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/stability" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stability</a></li></ul></section>Wed, 12 Jun 2013 00:27:43 +0000Matthew285 at http://marine.marsh-design.comhttp://marine.marsh-design.com/content/dynamic-stability-monohull-beam-sea#commentsUnderstanding monohull sailboat stability curves
http://marine.marsh-design.com/content/understanding-monohull-sailboat-stability-curves
<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>One of the first questions people ask when they discover I mess around with boat designs is: "How do you know it will float?"</p>
<p> </p>
<p>Well, making it float is just <a href="https://en.wikipedia.org/wiki/Archimedes%27_principle">Archimedes' principle</a> of buoyancy, which we all know about from elementary school: A floating boat displaces water equal to its own weight, and the water pushes upward on the boat with a force equal to its weight. What people usually mean when they ask "How do you know it will float" is really "How do you know it will float upright?"</p>
<p> </p>
<p>That's a little bit more complicated, but it's something every skipper and potential boat buyer should understand, at least conceptually. (Warning: High school mathematics is necessary for today's article.)</p>
<p> </p>
<h2>A yacht at an angle of heel</h2>
<p> </p>
<p>Let's consider a boat at rest, sitting level in calm water. The boat's mass is centred on a point G, the centre of gravity, and we can think of the force of gravity as acting straight down through this point. The centroid of the boat's underwater volume is called B, the centre of buoyancy. The force of buoyancy is directed straight up through this point.</p>
<p> </p>
<p>We now heel the boat over by an angle "phi". Point G doesn't move, but point B does: by heeling the boat, we've lifted her windward side out of the water and immersed her leeward side. The centre of buoyancy, B, therefore shifts to leeward.</p>
<p> </p>
<p>The force of buoyancy, acting upward through B, is now offset from the force of gravity, acting downward through G. The perpendicular distance between these two forces, which by convention we call GZ, can be thought of as the length of the lever that the buoyancy force is using to try to bring the boat upright. GZ is the "righting arm".</p>
<p> </p>
<p>If we draw a line straight upward from B, it will intersect the ship's centreline at a point called M, known as the "metacentre". (Strictly speaking, the term "metacentre" applies only when phi is very tiny, but a pseudo-metacentre exists at any given angle of heel.) The metacentric height is a useful quantity to know when calculating changes in trim and heel.</p>
<p> </p>
<p><img alt="Sailboat's midship section, heeled, with key points K, B, G, M, B(phi), Z marked." src="/sites/default/files/u4/2012/2012June/stability/righting_arm.svg" style="width: 560px; height: 560px;" /></p>
<p> </p>
<p>(Can't see the images? <a href="/sites/default/files/u4/2012/2012June/stability/diagrams_for_crappy_browsers.png">Click here for now</a>, then go update your web browser.)</p>
<p> </p>
<p>We can easily draw a few conclusions simply by looking at the geometry:</p>
<p> </p>
<ul>
<li>The boat will be harder to heel, i.e. more stable, if GZ is increased.</li>
<p> </p>
<li>Lowering the centre of gravity, G, will increase GZ.</li>
<p> </p>
<li>Moving the heeled centre of buoyancy to leeward will increase GZ.</li>
<p> </p>
<li>If GZ changes direction- i.e. if Z is to the left of G- the lever arm will help to capsize the boat instead of righting it.</li>
<p>
</p></ul>
<h2>Stability Curves: GZ at all angles of heel</h2>
<p> </p>
<p>To prepare a stability curve, the designer must find GZ for each angle of heel. To do this, she must compute the location of B at each angle of heel, and determine the height of G above the base of the keel (the distance KG).</p>
<p> </p>
<p>In the early 20th century, finding B at each angle of heel was an extremely tedious process involving a lot of trial-and-error, a lot of calculus, and days or weeks of an engineer's time. Today, this can be computerized, and takes only a few seconds once the hull is modelled in a CAD program. Finding KG, though, is still a tedious process: it can either be measured by moving weights around on an existing boat and measuring the resulting angle of heel, or it can be calculated by tallying up every piece of structure, ballast, equipment and cargo on the boat.</p>
<p> </p>
<p>Once that math is done, the designer can plot GZ (or righting moment, i.e. displacement times GZ) over all possible angles of heel. This produces the familar stability curve:</p>
<p> </p>
<p> </p>
<p> </p>
<p><img alt="Righting arm (GZ) curve showing regions of positive and negative stability, maximum and zero stability points, and typical sailing range." src="/sites/default/files/u4/2012/2012June/stability/gz_curve.svg" style="width: 560px; height: 460px;" /></p>
<p> </p>
<p>All yacht skippers should be at least somewhat familiar with their own boat's stability curve, and it's a useful thing to study when buying a boat. To read the curve, we look at the following features:</p>
<p> </p>
<ul>
<li>The <strong>slope of the curve at low angles of heel</strong> tells us whether the boat is tender (shallow slope) or stiff (steep slope).</li>
<p> </p>
<li>The <strong>righting moment at 15 to 30 degrees of heel</strong> tells us about the boat's sail-carrying power. A large righting moment indicates a boat that can fly a lot of sail; a boat with a lower righting moment will need her sails reefed down earlier.</li>
<p> </p>
<li>The <strong>maximum righting arm</strong> (or righting moment), and the heel angle at that point, tells us where the boat will be fighting her hardest to get back upright. If this is at a low angle of heel, we have a boat with high initial stability- she'll feel very stable under normal conditions, but a bit touchy at her limits, and relies on her skipper's skill to avoid knock-downs. If the maximum righting arm occurs at a very large angle of heel, the designer chose to emphasize ultimate stability- she'll be hard to capsize, but will heel more than you might expect in normal sailing.</li>
<p> </p>
<li>The<strong> angle of vanishing stability</strong> is the point where the boat says "Meh, I'm done" and stops trying to right herself. Looking at the diagram above, this means that Z is now at the same point as G. A larger AVS indicates a boat that's harder to capsize.</li>
<p> </p>
<li>The <strong>region of positive stability</strong> is the region in which the boat will try to right herself. The integral of the righting moment curve (i.e. the area of the green region) is an indicator of how much energy is needed to capsize her.</li>
<p> </p>
<li>In the <strong>region of negative stability</strong>, the boat will give up and roll on her back, her keel pointing skyward. The integral of this region (i.e. the blue area) tells us how much energy it'll take to right her from a capsize; if this area is relatively small, the waves that helped capsize her might have enough energy to bring her back upright.</li>
<p>
</p></ul>
<h2>Try it on a real boat</h2>
<p> </p>
<p>How does this apply to some real boats? Let's consider a 10 metre, 8 tonne double-ender yacht of fairly typical layout and proportions. The parent hull looks something like this:</p>
<p> </p>
<p><img alt="Perspective view of a 10 metre, 8 tonne generic double-ender sailboat hull with a medium fin keel." src="/sites/default/files/u4/2012/2012June/stability/base_hull.png" style="width: 560px; height: 400px;" /></p>
<p> </p>
<p>Keeping her draught (1.5 m), displacement (8 tonnes), length (10 m), freeboard, deckhouse shape, etc. the same, we'll adjust the shape of the midship section to yield four boats that are directly comparable in all respects except beam and section shape. Hull A is a deep <a href="http://intheboatshed.net/2012/02/18/jeff-stobbes-striking-and-beautiful-victorian-style-plank-on-edge-yacht/">"plank on edge" style</a>, hulls B and C are moderate cruising yacht shapes, and the wide, shallow-bilged hull D resembles an <a href="https://www.mysticseaport.org/">old sandbagger</a>- or a modern racing sloop.</p>
<p> </p>
<p><img alt="Midship sections for four related hulls, A through D, ranging from narrow to very beamy." src="/sites/default/files/u4/2012/2012June/stability/midship_sections.svg" style="width: 560px; height: 440px;" /></p>
<p> </p>
<p>Now, assuming that G lies on the waterline (so KG = 1.5 m), we can compute the righting arm GZ as a function of the heel angle. If we multiply the righting arm GZ by the displacement, we get the righting moment.</p>
<p> </p>
<p><img alt="Righting moments for hulls A through D if the centre of gravity is on the waterline in all four ships." src="/sites/default/files/u4/2012/2012June/stability/righting_moments_kg150cm.png" style="width: 480px; height: 289px;" /></p>
<p> </p>
<p>Some immediate observations from this graph:</p>
<p> </p>
<ul>
<li>The narrow hull "A" has relatively little sail-carrying power at low angles of heel, but will self-right from any capsize. Her good "ultimate stability" comes from using ballast to get G as low as possible.</li>
<p> </p>
<li>The wide hull "D" can fly a lot more sail, but if she goes over, she ain't coming back up. She gets her high "initial stability" from her wide beam, which moves the heeled centre of buoyancy farther to leeward.</li>
<p>
</p></ul>
<p>There's a problem, though: We've assumed an identical centre of gravity for all four boats. That's not realistic. The deep, narrow hull will have her engine and tanks low in the bilge; the wide hull must mount these heavy components higher up. Let's reduce hull A's KG measurement to 1.35 m, and increase hull D's KG measurement to 1.65 m, a more realistic value. We'll scale KG for the other two accordingly.</p>
<p> </p>
<p><img alt="Righting moments for hulls A through D with centre of gravity adjusted to true values (lower in deep narrow boat, higher in shallow wide boat)." src="/sites/default/files/u4/2012/2012June/stability/righting_moments_kgtrue.png" style="width: 480px; height: 289px;" /></p>
<p> </p>
<p>The overall conclusions don't change much, but we now have some realistic numbers to play with.</p>
<p> </p>
<ul>
<li>Hull A, the narrow one, will have a hard time flying much sail. She'll heel way over in a breeze. But she can't get stuck upside down.</li>
<p> </p>
<li>Hull B, a moderately slender cruising shape, also can't get stuck upside down- her AVS is 170 degrees. Her extra beam causes the centre of buoyancy to move farther to leeward when she heels, so she has more initial / form stability than hull A and can carry more sail.</li>
<p> </p>
<li>Hull C, which is typical of modern cruising yachts, has over twice the sail-carrying power of the slender hull A. She'll heel less, and since her midship section is much larger, she'll have more space for accommodations. The penalty is an AVS of 130 degrees. That's high enough that she can't be knocked down by wind alone, but wind plus a breaking wave- such as in <a href="http://www.nytimes.com/1988/07/11/sports/guide-to-avoid-broaching.html">a broach situation</a>- could leave the boat upside down until a sufficiently large wave comes along.</li>
<p> </p>
<li>Hull D, the broad-beamed flyer, can hoist more than three times the sail of hull A at the same angle of heel. She'll be quite a sight on the race course with all that canvas flying. Her maximum righting moment, though, is only 37% more than hull A's, which leaves less of a margin for error- hull D is more likely to get caught with too much sail up, and will reach zero stability at a lower angle of heel. If she does go over, she has considerable negative stability, making it unlikely that she'll get back upright.</li>
<p>
</p></ul>
<h3>Work to capsize</h3>
<p> </p>
<p>If you're one of that slim percentage who paid attention in high school physics, you're probably looking at those curves and thinking: "Force (or moment) as a function of distance (or angle).... hey, <a href="https://en.wikipedia.org/wiki/Work_%28physics%29#Torque_and_rotation">if you integrate that, you get the work done</a>!</p>
<p> </p>
<p>And so you do, with the caveat that we're using a static approximation to a dynamic situation. The results are valid for comparison, but the actual numbers may not mean very much.</p>
<p> </p>
<p>Let's do that for each of our hulls. We'll integrate the righting moment curve as a function of heel angle, up to the angle of vanishing stability, to get the work done to capsize the boat. We'll also integrate from the AVS to 180 degrees to get the work done to right the boat from a capsize.</p>
<p> </p>
<p><img alt="Energy needed to capsize and to right each of the four hulls. The energy to capsize is similar (5500 to 6000) for all four; only the widest hulls require energy (~2000) to right from a capsize" src="/sites/default/files/u4/2012/2012June/stability/capsize_energy.png" style="width: 480px; height: 289px;" /></p>
<p> </p>
<p>Our four boats require roughly the same work to capsize! Changing the shape of the midsection affected the <em>shape</em> of the stability curve- a wider boat had more initial stability and less ultimate stability. In this case, though, our vessels are all about the same size and require about the same amount of work to capsize.</p>
<p> </p>
<p>Righting from a capsize is another matter. The narrow, deep hulls A and B will self-right without any outside influence- a nice confidence-booster if you're heading into the open ocean, although the reduced sail-carrying power and limited interior space of these vessels will probably be more important to most skippers.</p>
<p> </p>
<p>The moderate cruising hull, C, needs a bit of help to self-right, but any combination of wind and waves that can do 95 kN.m.rad of work on the boat is likely to produce a wave that can do 10 kN.m.rad of work on that same boat.</p>
<p> </p>
<p>Our broad-beamed racer, hull D, is not so fortunate. Righting her from a capsize takes one-third the work that capsizing her in the first place did, and her acres of canvas were probably a major factor in the initial capsize- they're now underwater, damping her roll motion instead of catching the wind. The odds are that this boat will stay upside-down until someone comes along with a tugboat or crane.</p>
<p> </p>
<h2>Lessons Learned</h2>
<p> </p>
<p>What's the take-home message from all this?</p>
<p> </p>
<p>If you're buying a new boat: Look at her stability curve, and compare it to other boats.</p>
<p> </p>
<ul>
<li>Good: Large region of positive stability, small region of negative stability, high angle of vanishing stability, steep slope at low heel angles.</li>
<p> </p>
<li>Iffy: Shallow slope at low heel angles (makes it hard to fly lots of sail, excessive heeling when underway).</li>
<p> </p>
<li>Risky: Low angle of vanishing stability, large region of negative stability.</li>
<p>
</p></ul>
<p>If you already have a boat:</p>
<p> </p>
<ul>
<li>Use the boat's stability curve to help avoid, and plan for, emergencies.
<ul>
<li>If you know her point of maximum stability, you can be sure to reef the sails well before that point.</li>
<p> </p>
<li>If you know her AVS and the shape of the curve in that region, then when a broach or knockdown happens, you already know how hard she'll fight to come back upright.</li>
<p> </p>
<li>If you know how much area is covered by the negative stability region of the curve, you'll have some idea of whether she'll come back from a capsize on her own or else have to wait for help.</li>
<p>
</p></ul>
</li>
<p> </p>
<li>Know the location of the designer's intended centre of gravity (point G).
<ul>
<li>Determine if anything you've changed- a dinghy added on the deck, perhaps- has moved the centre of gravity.</li>
<p> </p>
<li>If G has moved, adjust your mental model of the stability curve accordingly: just shift the curve up or down by (change in height KG) * sin(heel angle).</li>
<p>
</p></ul>
</li>
<p>
</p></ul>
<h2>Confounding Factors</h2>
<p> </p>
<p>What we've discussed here is just about how to read the stability curve- it's not a complete picture.</p>
<p> </p>
<p>There are many other factors that must be considered to get a complete understanding of a boat's stability. Among them:</p>
<p> </p>
<ul>
<li>Dynamic effects. Everything discussed so far is for the static case, and is good for comparison purposes. But in practice, boats move.</li>
<p> </p>
<li>Waves. Stability curves are calculated for flat water, ignoring the effect of waves.</li>
<p> </p>
<li>Differences in rigging. Weight aloft has a much larger effect on the boat than weight down low- particularly where the roll moment of inertia, an important property for dynamic stability, is concerned.</li>
<p> </p>
<li>Keel shape. Keels tend to damp rolling motion; this behaviour is quite different with a long keel than with a fin keel, or with a fin keel underway versus a fin keel at rest.</li>
<p> </p>
<li>Downflooding. Everything we've discussed here assumes that the boat is watertight in any position. If she takes on water when rolled, everything changes.</li>
<p> </p>
<li>Cockpits. Our demonstration boat doesn't have a cockpit. A large cockpit could hold several tonnes of water- and with a free surface, no less. That means that G will move all over the place, usually in the wrong direction.</li>
<p>
</p></ul>
<h2>Further Reading</h2>
<p> </p>
<p>Steve Dashew's article "<a href="http://www.setsail.com/evaluating-stability-and-capsize-risks-for-yachts/">Evaluating Stability and Capsize Risks For Yachts</a>", and others on his site, discuss stability-related risks as they relate to cruising yachts.</p>
<p> </p>
<p>Technically-minded readers should refer to a naval architecture textbook, of which my present favourite is Larsson & Eliasson "Principles of Yacht Design" (McGraw-Hill).</p>
<p> </p>
<p>Don't even think about buying a cruising yacht without first reading <a href="https://www.morganscloud.com/category/boat-design-selection/bds-articles/">John Harries' extensive series of articles on boat and gear selection</a>.</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/stability" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stability</a></li><li class="field-item odd"><a href="/category/boats/monohull" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">monohull</a></li><li class="field-item even"><a href="/category/boats/sail" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">sail</a></li></ul></section>Sat, 09 Jun 2012 13:57:39 +0000Matthew120 at http://marine.marsh-design.comhttp://marine.marsh-design.com/content/understanding-monohull-sailboat-stability-curves#comments