Dynamic Stability of a Monohull in a Beam Sea

The last post in our series on yacht stability looked at the static case. 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.

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.

Warning: Math!

Our earlier introduction to static stability was relatively light on mathematics. While some calculus is required to produce a stability curve, reading it requires only a conceptual understanding, not any actual calculations.

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.

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.

I, the Moment of Inertia

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".

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.

We can state that more generally and more formally as

$I = m r^{2}$

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:

$I = \sum\limits_{i=1}^{N} m_{i} r_{i}^2$

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:

$I = \int\limits_V \rho(\vec{r}) \vec{r}^2 dV$

In other words, the moment of inertia increases if:

  • The mass increases (twice the mass, twice the moment), or
  • The mass is placed farther from the centre of rotation (twice as far, four times the moment).

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.

Angular Acceleration

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:

$\vec{\tau} = \vec{r} \times \vec{F}$

The second relates the torque $\vec{\tau}$ to the acceleration $\vec{\alpha}$, and the relationship is- surprise, surprise- the moment of inertia.

$\vec{\tau} = I \vec{\alpha}$

Enough equations, let's look at a boat

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:

  • $F_{sail}$, 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.)
  • $mg$, the force of gravity on the boat (equal to the boat's mass times the gravitational acceleration).
  • $F_b$, 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.
  • $F_{lat}$, the laterally-directed lift of the keel, rudder and underbody as they move through the water.

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.

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.

Top o' the Wave

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.

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/s2, we perceive the force of gravity to be zero.

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.

Here's our first big lesson in dynamic stability: Buoyancy is based on weight, not on mass. Our boat's mass hasn't changed, but when we go into free-fall on the crest of the wave, the boat's weight 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.

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/s2), we have this:

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.

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

$\vec{\tau} = a F_{lat} + b F_{sail}$

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:

$\vec{\alpha} = \frac{\vec{\tau}}{I}$

So a higher torque or a lower 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.

Changing the Yacht's Behaviour

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.

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).

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 static stability is slightly improved by losing all that weight up top, her dynamic stability is severely compromised; without the mast, she'll be much more susceptible to wave-induced capsize.

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.

Our other tool for improving this situation is to reduce the net overturning torque on the yacht.

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.

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.

Skidding Sideways

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.

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.

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.


We may summarize this article in a few take-home points:

  • The dynamic behaviour of a yacht in waves is at least as important, if not more so, as its static stability curve.
  • When we're taking heavy weather from abeam, we want to keep roll accelerations low.
  • Roll accelerations are reduced if we have a high roll moment of inertia or if we reduce the overturning forces.
  • 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.
  • 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.
  • All of this is less of a problem if we can keep the boat bow-on to, or running with, the waves.

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.




Thank you

for that explanation/ you would make a great teacher, I actually understand it now.

Great Explanation

Wonderfully simple explanation, but the math is cool too. Thank you.

Just asking

The text box in the second diagram states "after crest of wave", implying the wave moves from the right to the left. The arrow depicting the force on the sail by the wind points from the left to the right.
Is your analysis about wind generated waves? If so, how can the wave be running into the wind?

Wave sources

Matthew's picture

In the ideal case, wind and waves are coming from the same direction.
In practice, that's often not true. You frequently get a small-amplitude, short-wavelength wind-driven wave system superimposed on a much larger amplitude, longer wavelength system of waves from a distant origin. Or you get gusts, turbulence, changes of wind direction, wind opposing current to whip the waves up even sharper.... all told, it's never safe to assume that the wind direction and the dominant wave direction are related.

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