It seems like that the greater the surface area below the waterline, the greater the resistence to being a roly boat in a roly anchorage; or roly underway in the case of a cross-swell. Mass would also be a factor, so a heavy-displacement, full-keel boat *should" be the most reisistant to roly-ness and thus the most comfortable boat in a cross-well anchorage or underway with a cross-swell. This is important, as both my wife and I find that - for us - roll is the largest contributor to motion discomfort. (Note that I"m making a distintion between roll and heel.)

I understand why intuitively you might think that a heavy displacement- Full keel boat would be less rolly than a lighter weight boat but that is rarely the case, at least if we define being rolly as rolling through a large roll angle. For a wide variety of reasons, heavy displacement- full keeled boats tend to roll through wider angles, but at slower speeds. The following is a draft article that I wrote some years ago which has a semi-detailed explanation of the factors impacting motion. I apologize that it is quite long and not completely on topic.

Motion Comfort and Stability basics- a detailed discussion

To begin this discussion I think that it is important to define the characteristics that would produce a boat with a comfortable motion. In a general sense, a boat design that had desirable motion comfort characteristics would be a boat met some basic criteria in terms of balancing the shape of the boat (buoyancy distribution) and the weight distribution within the boat so that the boat minimized both the amount of motion and amount of the acceleration/ de-acceleration felt by the crew.

The amount and rate of change in speed of motion a boat experiences is related to the amount of energy that is imparted into the boat, the ability of the boat to store that energy, and the ability of the boat to dampen (disburse) that stored energy. A boat with a comfortable motion will minimize the amount of force generated, or attenuate the length of time that the energy is absorbed or dissipate that energy in some way other than by changing the rate of speed of the motion.

We often tend to look at boats using simple static calculations, but in the world of motion comfort or capsize, dynamic issues often come into play. For example, if you had a boat that has form stability, but which has absolutely no inertia, no roll moment of inertia or non-form (weight) stability, (which of course cannot actually exist in real life) that boat would always sit at the same angle relative to the surface of the water. In other words, no matter where that boat sat on a wave, its waterline would essentially be tangential to the surface of the water.

In that example the rate of change in the angle of the boat and the vertical rate of change would precisely follow the face of the wave passing below it. That boat might have reasonable motion comfort in flat water or small waves with a long duration, but would quickly become pretty uncomfortable in steeper waves or more closely spaced waves where roll, pitch and heave would toss the crew around mercilessly.

Of course if we took that same example and added some inertia (bringing it into the real world), the boat would no longer instantaneously change direction vertically or with regards to roll angle with the surface of the water. There would be a lag between the change in vertical height and angle of the wave face and the change in height and angle of the boat. In theory, this slowing in pitch and roll due to inertia would be more comfortable to the occupants since there would be less dramatic accelerations felt by the crew of the boat. It might also offer better performance because of less extreme interruptions to the wind and water flows over the foils.

In discussions of motion, motion is classified as being either linear (also referred to as directional) or rotational. Linear motion is described in one of three ways: surge (fore and aft), leeway (sideward, also referred to as sway), and heave (vertically). Rotational motion is also described in one of three ways: Pitch (fore and aft), yaw (rotation as seen in plan view) and roll (abeam, which of course is side to side seen from bow or stern). In all of these motions, acceleration is slowed by inertia. In the case of linear motion, the amount of inertia is basically proportionate to the weight of the boat, the amount of force encountered by the boat, and any other types of resistance that might dampen that force (i.e. in the case of surge, the drag of the boat would reduce the acceleration due to the force of a wave from astern).

In the case of heave, the speed of vertical upward acceleration is relative to the rate at which there a is vertical change in height of the wave face, and is also is proportionate to the weight of the boat as well as also proportionate to the area of the water plane (the area of the boat measured at the surface of the water) of the boat.

In other words in heave, going up a wave, a heavier boat will accelerate proportionately more slowly than a lighter boat with an equal waterplane. The waterplane comes into play on the acceleration of the upward climb since the force imparted into the boat is roughly equal to the plan dimension of the boat. If we compare two boats of equal waterplane, the boat with the smaller water plane will sink lower into the wave than the boat with the larger waterplane. That small amount of reduction in the initial travel distance as the boat lifts results in the boat with the smaller waterplane having slightly less acceleration in effect acting as shock absorber. This effect is more noticeable in closely spaced waves.

Several factors impact the area waterplane of the boat, but in general, the lighter the boat and the deeper the weighted average depth of canoe body, the smaller the water plane. In a general sense, since the amount of heave is proportionate to the mass of the boat and the areas of its water plane, a lighter boat with a proportionately smaller waterplane (proportionately deeper cane body) may actually have a more comfortable heave motion than a heavier boat with a proportionately larger water plane.

At the top of the wave, the boat's inertia will carry it vertically beyond past the point that the wave is trying to lift the boat. Normally the carry is only a small distance but in extreme cases; a steep wave with lots of speed, a boat with a lot of inertia and a large water plane, the boat can be thrown clear of the top of the wave. In this case it is possible for the boat to get out of phase with the back of the wave and impacting quite solidly and uncomfortably.

Roll

Harmonic motion- think about the dynamics of roll as it pertains to motion comfort, we can look at two cases and see how the placement of the weight impacts the rotational motion of the boat. Let's look at two boats with equal height masts and equal depth and profile keels, and equal hull shapes. But one of these boats is constructed conventionally and the other with higher tech construction techniques. For the sake of discussion let us assume that that the low tech boat has the same 40 mast as the high tech boat, and for the sake of discussion, we put a 100 lb weight at the top of its mast. That weight would increase the roll moment of inertia by 160,000 pound-feet squared (100 lbs x 40 ft x 40ft) and reduce the roll rate of the boat. The other boat does not have the weight at the top of the mast but neither does it have the teak decks, or heavy teak interior, or a hull liner, or marble counter tops. Instead it has a 3,300 lb bulb in the bottom of its keel, the center of gravity of that bulb being 7 feet below the roll center of that boat. In this case we have 3,300lb x 7 ft x 7 ft or an increase in roll moment of inertia of 161,700 pound-feet squared, which is similar to the increase in the moment of inertia for the 100 lb weight at the mast head.

I think that we would all agree that both should have a slower harmonic motion than an equal weight boat that did not have either the weight at the mast or bulb at the bottom of the keel. But despite the fact that these boats have equal roll moments of inertia, I suggest that the behavior of the boat with the weight at its masthead as compared to the boat with the bulb will be extremely different in a seaway.

If we look at mechanics of a boat lying beam to the waves and rolling in steep seaway, 1. there will be a rotational moment that is generated by the change in angle of plane of the wave face acting against the form stability of the boat, and 2. another rotational moment that comes from sheer of the surface of the wave acting on the underbody of the boat that results from the difference in the speed of the water at the surface which is moving faster relative to the water deeper in the wave, and 3. a third rotational moment that comes from gravity acting on the boat causing it to want to slide down the inclined plane of the wave and but that slide is being resisted by the keel creating a moment between the center of actual resistance and the center of the momentum of the boat.

In our example of the two boats, since they have the same hull, weight and underbody profile, it would suggest that they will feel a similar roll inducing moment due to moment #1 and moment #2.

But when we look at roll inducing moment #3, in this case the boat with the weight at the top of its mast will have a higher vertical center of gravity and so the lever arm between the force couple (lateral resistance and center of gravity) will be longer creating a greater rotational force, and since both boats have an equal roll moment of inertia, there will be a faster roll rate for the boat with the mast weight than the keel weight.

But also, if we continue to look at the rotational moments acting on the boats, in the case of the boat with the keel weight, the weight of the keel is acting in couple with the center of buoyancy creating a moment trying to right the boat vs. the weight at the top of the mast, which is also creating a couple with the center of buoyancy but one trying to overturn the boat, and so with equal inertia would tend to further increase the roll rate of the boat with mast weight.

If we think then of these two boats sliding down the face of the wave an into the trough, when they hit the bottom of the wave, the boats being of equal weight and profile will hit the bottom of the wave moving at the same lateral velocity, but the boat with the mast weight will hit with a greater rotational speed.

If we look at the forces felt by the crew on the boat, we would need to look at the rate of deceleration. As I assume that we would agree upon, the force of deceleration will be proportionate to the change in speed, and the distance/time over, which that change of speed occurs. To some extent we do not have enough information, but in a general sense, boats in principle would experience an equal form stability righting moment at the bottom of the wave if they were at an equal heel angle.

But since the boat with the mast weight has a greater rotational speed, it will actually have a greater heel angle at the bottom of the wave. In theory this greater heel angle would be result in a higher rate of side motion since the keel will have rotated out of the water flow but we can ignore that for a moment.

Since the boat with the mast weight has a greater rotational speed at the trough and it will need experience a greater change in speed as it flattens out to begin its climb. That greater speed will create a greater momentum. Assuming an equal hull form creating the righting moment for both boats, this greater momentum relative to righting moment will result in the boat with the mast weight having a higher heel angle at the time that the boat stops moving in the trough and begins rotate back to level. This greater momentum would also generate a greater impact force which would be felt by the crew as well. In other words the boat with the mast weight would likely experience a less comfortable motion having some mix of greater impact force and a larger roll angle experienced by the crew.

By the same token, looking at likelihood of a capsize, as has been discussed earlier, as boats heel there is a point of maximum righting moment after which righting moment decreases. The amount of that maximum righting moment and the speed at which it decreases is mostly dependant on the hull/cabin shape and weight distribution.

Continuing our example, the boat with the mast weight will be at a higher angle of heel when it impacts the trough of the wave. It will also have a reduced limit of positive stability that would result from having a higher vertical center of gravity as compared to the boat with the keel weight. So you have a boat that is rotating at a higher speed, hitting the bottom of the wave at a larger heel angle that has a smaller LPS, and a more rapid loss in righting moment as it approaches its LPS.

I would think that we would all agree that assuming the two boats in our example have equal moments of inertia, the boat with the mast weight would be more likely to capsize than the boat with the keel weight. I would think that this example provides one small case where weight distribution can be shown to be a significant determinant in both motion comfort and the likelihood of capsize.

As mentioned above one of the core factors in reducing both the roll rate and the roll angle is dampening. The majority of the dampening comes in the form of the force required to rotate the keel, and rudder sidewards through the water and the sail plan sidewards through the air. Again this is a moment of inertia problem so that the area of the sails and keel count linearly while the distance from the instantaneous roll axis to the centroid of the dampening forces are exponential. In other words a shallower draft keel needs a lot more area to equal the dampening of a deeper draft keel. Similarly a short rig, needs a lot more area to equal the dampening of a taller rig.

Additionally, long length, shallower draft keels are often associated with boat which are heavy for their length. Boats which are heavy for their length tend to have deeper cane bodies and so while they may have a lot of area when seen in profile, a smaller proportion of this area is actually a vertical face of the keel capable of generating dampening forces. This combination of proportionately smaller area with a smaller dampening moment of intertia is the reason why deep draft fin keel boats often have better roll charteristics than a boat with longer shallower keel.

Further contributing to this is a reduction in the dampening force per unit area. When rolling, the portion of the keel near the hull generates less dampening force than portions of the keel which are further than the hull. In the case of shallower, longer keels, more of the keel is operating in an area of water that is closer to the hull. To explain, as a hull rotates it creates turbulence and moves some of the water with it. Because a larger percentage of a long shallower draft keel is operating in this area of turbulent and moving water, the forces on any square area of the keel near the hull will be lower than an area of a deeper keel whose areas are further from the hull and further out of this zone of turbulent and moving water.