Designing boat hull shapes for stability involves a number of trade-offs. Not least is the comfort of the crew and passengers. Not only must our

Figure 5.8. The radius of gyration (dashed circles) about the center of gravity, CG (small circles), of a boat hull for different hull shapes. If the hull is a hollow cylinder (a), the radius of gyration is just the cylinder radius. For a solid cylinder (b) the radius of gyration is only 71% of the cylinder radius. For different boat hull cross sections (c)-(f) the radius of gyration is different, as shown.

Figure 5.8. The radius of gyration (dashed circles) about the center of gravity, CG (small circles), of a boat hull for different hull shapes. If the hull is a hollow cylinder (a), the radius of gyration is just the cylinder radius. For a solid cylinder (b) the radius of gyration is only 71% of the cylinder radius. For different boat hull cross sections (c)-(f) the radius of gyration is different, as shown.

boat be stable, but she must be compatible with life aboard. A flat boat with a stiff response to heeling—one that rights herself sharply—may buffet those aboard, perhaps dangerously. Passengers may be knocked about as their yacht rolls back quickly to her upright position. On the other hand a tender response—from a boat that rights herself very slowly —may induce unease. If you are on board such a vessel, rolling slowly from side to side over large heeling angles, you may wonder whether or not she is ever going to come back or if she has passed the point of no return. Or you may simply be wondering whether or not you can hold down your breakfast. So the rolling behavior of a boat is important. Rolling is a consequence of stability, of course. We have seen how Puddleduck rights herself in response to heeling over. She has inertia, and the righting does not cease when she gets herself vertical. She overshoots and rolls the other way before coming back again. The resultant rolls will diminish in amplitude as hydrodynamic frictions damps out the motion (pardon the pun).* In this section I will investigate the rolling motion of a boat.

*Amplitude here means maximum heeling angle. So the maximum heeling angle is reduced with each roll due to friction.

First, I need to introduce yet another technical term.* Radius of gyration may sound to you like the size of a dance floor, but to an engineer it has a more prosaic meaning. Figure 5.8 depicts the profiles of a number of boat hulls and other floating objects. These objects may be induced to roll about their CG, and we need to know the effective radius of the object, as measured from the CG. For a hollow cylinder centered on the CG this is a no-brainer: the radius of gyration, Rg, is just the radius of the cylinder. For a solid cylinder of the same mass, however, it turns out that Rg is only 71% of the actual radius. This is because the cylinder mass is distributed over many radii, in this case from the CG (radius zero) out to the actual, physical cylinder radius, R. The "average" radius, when it comes to rotations, is Rg = 0.71R.4 For more complex shapes such as a boat hull, of which several are shown in figure 5.8, the radius of gyration is difficult to work out algebraically; we must resort to a computer. I don't intend to do that here; so long as you appreciate that there is a radius of gyration for any rotating object, then I have got the message across.

Now we will calculate the period of a roll (a.k.a. roll time) for the boat profile shown in figure 5.8f. This profile is a half-cylinder of radius R, and we may assume that it is a reasonable approximation to the shape and mass distribution of many open boats. Because we are dealing with simple, approximate solutions, rather than complex, exact simulation results, this half-cylinder shape will serve admirably for the purpose of calculating roll time. It has a couple of features that reduce the math to a manageable form. In figure 5.9 we have the half-cylinder boat rolling on a calm sea. The roll angle induces a torque which tends to right the boat. If she is heeling counterclockwise, as shown, the torque acts clockwise, and vice versa.

We invite Sir Isaac Newton to cast his expert eye over the scene, and he obliges us by quickly deriving the equation of motion, which tells us how the boat rolls. We can easily solve the equation, and determine the roll time,5 which is

* If you supposed that the plethora of technical terms to do with boat torque reflects the considerable amount of theoretical effort that has been put into understanding this subject, you would suppose right.

dmz dmz

Figure 5.9. For a half-cylinder hull shape on a flat sea, the center of buoyancy, CB (x), does not change with roll angle, amz, and the metacenter (triangle) is always at the cylinder axis. The buoyancy force (arrow) applies a torque about the center of gravity, CG (open circle), and from this force we can derive and solve the roll motion equation. The metacentric height is noted by hGM.

Figure 5.9. For a half-cylinder hull shape on a flat sea, the center of buoyancy, CB (x), does not change with roll angle, amz, and the metacenter (triangle) is always at the cylinder axis. The buoyancy force (arrow) applies a torque about the center of gravity, CG (open circle), and from this force we can derive and solve the roll motion equation. The metacentric height is noted by hGM.

Equation (5.2) tells us that the time it takes our half-cylinder boat to make one complete roll does not depend on the size of the roll: it can be 5° or 25°, but the roll time is the same. Roll time does increases with the size of our boat, which is no surprise. One rule of thumb used by boat designers is that the roll time for any boat should be pretty close to the beam, meaning that the roll time in seconds should be about equal to the beam dimension measured in meters. The reason for this rule is crew and passenger comfort. Faster roll times create jerky, harsh motion, while slower roll times can induce queasiness. For a comfortable boat ride the hull should be built with a metacentric height, hGM, and radius of gyration, Rg, to comply. For our half-cylinder we have Rg ~ hGM ~ 0.8R, where R is the physical radius of the boat. Plugging this radius of gyration into equation (5.2) gives us a roll time of 1.6 seconds for a boat radius of 0.8m (and so a beam of 2R = 1.6m). So, the requirement for a comfort able outing in our half-cylinder boat constrains the boat beam to be between 5 and 6 ft.

Real boats are more complicated to analyze, but the same kind of results apply. Designers must balance a number of different requirements to produce a good boat. The metacentric height and the radius of gyration are juggled with the desired boat beam to yield an acceptable roll time.

Modern boat designs can "cheat" equation (5.2) by introducing other influences. For example, a large keel will dampen down rolling motion, as we will see, so that any discomfort that results will be of brief duration. Some large and expensive boats have a keel that can be canted. It can be set out to one side or another to counter rolling by increasing the amount of righting torque. This is an expensive and not altogether satisfactory solution,* but it shows the lengths to which designers are prepared to go in order to reduce roll. Another solution that has been considered is to install movable ballast. Imagine a heavy weight that can be shifted dynamically from one side of your boat to the other, with the movement timed to provide maximum righting torque. Such a dynamic solution would require computer-controlled equipment for shifting the weight in a timely manner. An altogether simpler solution, in wide use, is to deploy flopper-stoppers—floats that dampen rolling motion. These work only when the boat is moored, but they cut out the big rolls that can result from, say, the wake of a large passing vessel striking you on the beam. Flopper-stoppers work by significantly increasing the damping effect of friction (fig. 5.10).

Puddleduck is in harbor one calm evening, and you are entertaining a lady friend on board. A large cruiser sails silently by. You pour a glass of red wine. The cruiser's wake hits Puddleduck on the beam, and you spill the wine over the lady friend. She is about to remonstrate about her ruined white dress when the next wave hits, tipping her onto the floor

* It is technically difficult to install an effective keel that moves in this way. Also, a canted keel is less effective at resisting leeway motion. So here we have another example of "trade-off": different design requirements tug boat designers in different directions. Good design is all about successful compromises.

Figure 5.10. Rolling motion with no damping (dashed line), more realistically with a little damping (thin line), and with flopper-stoppers deployed (bold line). These curves have been simulated—they are not experimental results obtained by measuring rolling boats, and the time axis is arbitrary—but they capture the physics of what is happening. The flopper-stoppers reduce the duration, as well as the peak amplitude, of the roll.

Figure 5.10. Rolling motion with no damping (dashed line), more realistically with a little damping (thin line), and with flopper-stoppers deployed (bold line). These curves have been simulated—they are not experimental results obtained by measuring rolling boats, and the time axis is arbitrary—but they capture the physics of what is happening. The flopper-stoppers reduce the duration, as well as the peak amplitude, of the roll.

and you on top of her. She is about to remonstrate about inappropriate behavior when the next wave rattles you both around Puddleduck like two peas in a can. She is about to leave when the next wave tips her overboard. OK, so my humorous introduction to rocking and rolling is also a little overboard, but I hope that you can appreciate the wave phenomenon epitomized here.

Your thoughts, on seeing your lady friend tipped overboard by a series of waves, as she shrieks for help and splashes frantically in the water, naturally turn to wave resonance. A series of waves, particularly if they approach a boat broadside on, can instigate rolling behavior that is much more severe than that induced by a single wave. The phenomenon of resonance is at work here. One boat in the harbor may rock violently back and forth, while none of the others seem much affected. It is like the wineglass (empty, this time) that is shattered by a diva's voice, but only if the pitch is just so.*

I can demonstrate this familiar wave resonance phenomenon to you by again making use of the half-cylinder boat. This time, as you can see in figure 5.11, I have given her a keel. The reason for doing so is to emphasize the role played by friction in the resonance phenomenon. As with flopper-stoppers, a keel helps to mitigate the effects of rolling motion by increasing the effective friction between the boat and the water. It is straightforward to set up a simple (approximate) mathematical model and solve the equation of motion for such a boat influenced by a series of waves. First, we model the waves by a sinusoid of specified wavelength and speed, and then let loose Sir Isaac once more to tell us how the boat's rolling angle changes when she is hit broadside by these waves.6 The answer is that the rolling angle changes in time as follows:

A transient is a disturbance that settles down quickly, leaving a more persistent solution. I will ignore the transient behavior here. What is left, once the transient effects have dissipated, is a rocking motion with angular frequency, v and amplitude, a0.t The quantity 9 is a phase factor, which describes by how much the rocking angle of the boat lags behind the water wave. This phase angle is predicted by my model but is not of much interest here, so let's ignore it. The angular frequency and amplitude can be expressed in terms of water wavelength and boat parameters:

Here A is water wavelength, b is friction coefficient as before, and v V A ' a" V(œ0- vO2 + bfv ' (5-4)

We have already met radius of gyration, Rg. The CG and CB distances hCG, hCB are constant for our half-cylinder boat and are defined in figure 5.11. Water wave height is h.

*At this point you dive in, rescue lady friend, send her tearfully home in a cab, and return to thinking about wave resonance.

tAngular frequency is frequency divided by 2p, so an angular frequency of 6.28 sec-1 is the same as 1 Hz, or 1 cycle per second.

Figure 5.11. For a half-cylinder hull hit by a broadside wave, the buoyancy torque is as shown by the arrow. The righting arm, GZ, is the horizontal distance between the center of gravity, CG (open circle), and the center of buoyancy, CB (x). Metacentric height is hGM. Roll angle is amz, and wave slope angle is aCB. Note that the metacenter lies below the CG, and so the torque in this instance tends to capsize the boat. Note also the distinction between CG distance, hCG, and metacentric height, hGM; they were the same in fig. 5.9, but not here. Inset: A short time later the metacenter is above the CG, and so the torque acts to right the boat. The metacenter moves wildly, resulting in uncomfortable rolling motion.

Figure 5.11. For a half-cylinder hull hit by a broadside wave, the buoyancy torque is as shown by the arrow. The righting arm, GZ, is the horizontal distance between the center of gravity, CG (open circle), and the center of buoyancy, CB (x). Metacentric height is hGM. Roll angle is amz, and wave slope angle is aCB. Note that the metacenter lies below the CG, and so the torque in this instance tends to capsize the boat. Note also the distinction between CG distance, hCG, and metacentric height, hGM; they were the same in fig. 5.9, but not here. Inset: A short time later the metacenter is above the CG, and so the torque acts to right the boat. The metacenter moves wildly, resulting in uncomfortable rolling motion.

So what do these equations tell us about boat rocking due to a series of waves on the beam? Any physicist looking at the form of equation (5.4) for amplitude a0 would immediately cry out "resonance!" The amplitude can vary over an enormous range depending on boat and wave characteristics. In detail:

As you would expect, the boat rocks at the wave frequency.

->The rocking amplitude is at a maximum when = v, which is to say when the water wavelength is l = 2pR;/hCG. When the wavelength equals this combination of boat hull gyration radius and CG height, the boat resonates with the wave, producing maximum rocking action. If the boat hull parameters are different, rocking amplitude is less. This difference in hulls explains why some boats are affected more than others by the same set of waves. •—'Given that resonance occurs, the amplitude of the boat roll is found to equal the following messy expression: a0max = VghCGhCBh/bR3g. Let me try to simplify this a little. We expect that hCG and hCB are both proportional to Rg, and so we can say that the maximum roll amplitude behaves like a0max ~ h/bR3g2. Bigger waves mean bigger rolls, unsurprisingly. Increasing friction helps reduce roll amplitude, so deploy those flopper-stoppers. Most significantly, increasing the radius of gyration significantly reduces roll amplitude. Thus, bigger boats suffer much less than smaller ones.

Everything I have said in this section applies to monohulls. We can perform a similar calculation for catamarans, and as you would expect, the results are rather different. This section is already technical enough, so I won't extend it with yet more math but will simply summarize the outcome. We obtain resonance behavior for catamarans in the same way as for monohulls, but the peak amplitude is smaller in general because catamarans are wider. Indeed, if the wavelength happens to equal the catamaran hull separation, there is no rolling motion at all; the cat just rides up and down with the waves. The conditions for resonance, when it does occur, are very different from the monohull case. Here, the key factor is the average hull depth, d0, under water. Resonance happens when the water wavelength is k~ 2pd0. So we see that a flotilla of cats at anchor will respond differently to the wake of a passing cruiser. Some will agitate violently, and other will not.

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