Myboatplans 518 Boat Plans

Torque applied about the longitudinal axis leads to heeling, and if the heeling angle is large enough, a boat may capsize. This is considered to be undesirable among all the boat owners I know, and so hull stability is taken to be a matter of some importance. Here, stability refers to the ability of a boat to right itself after having been heeled over by, say, a sudden gust of wind or a sudden change of course. We would like to know how much our boat can heel over before capsizing, and we would like to know how fast she can right herself. I will deal with these questions here. The physics of hull stability is simple in concept but devilish fiddly when it comes to detailed calculations. So, in the spirit advocated in the introduction, I will once again simplify when necessary to make the issues clearer and will relegate math details to the endnotes. The fiddly details have generated much literature, and a number of technical terms have entered the sailor's vocabulary as a consequence. So that you know what is being talked about when these words are uttered, I will begin by introducing them.

The force of gravity that acts on an extended object, such as your Pud-dleduck or indeed yourself, can be thought of as acting at a particular point, called your center of gravity (CG). This is a familiar idea, and it extends to other forces. Thus, the force that buoys up your boat acts on all parts of the wetted surface but can be thought of as acting at a single point called the center of buoyancy (CB). So far, so good, but as soon as we think about CB a little more deeply, we see that it quickly becomes complicated. Consider figure 5.5. Here, for simplicity I have replaced your beloved Puddleduck with a block of wood—no insult intended— but you will appreciate the reason for this substitution shortly. The block of wood is of such a density that the water line is halfway up when the block is floating in a stable position. The CG is easy to locate: it is simply the geometrical center of the block, assuming uniform density. What about the CB?

The center of buoyancy for a simple shape such as this block is the geometrical center of that portion of the block which is underwater. So, for the initial stable position we can readily mark the CB with an x, as in figure 5.5. Now let us suppose that the block is disturbed and rotates a

x |

Figure 5.5. (a) A floating block in a position of stable equilibrium. The center of gravity, CG, lies in the same vertical plane as center of buoyancy, CB (x). (b) When the box is rotated, its CB moves and is no longer in the same plane as the center of gravity, resulting in a restoring torque. (c) and (d) For increased rotation angles, the CB changes, thus altering the torque that is applied to the box. (e) When the rotation angle is 90° the CG and CB are again aligned and no torque applies. But this position is one of unstable equilibrium.

Figure 5.5. (a) A floating block in a position of stable equilibrium. The center of gravity, CG, lies in the same vertical plane as center of buoyancy, CB (x). (b) When the box is rotated, its CB moves and is no longer in the same plane as the center of gravity, resulting in a restoring torque. (c) and (d) For increased rotation angles, the CB changes, thus altering the torque that is applied to the box. (e) When the rotation angle is 90° the CG and CB are again aligned and no torque applies. But this position is one of unstable equilibrium.

little, like a heeling hull. This new orientation does not alter the CG at all, but the CB has moved. We can calculate the new CB easily enough for a simple block of wood, but for a complex hull shape it can be impossible to work out with a pen and paper: we would need to resort to number-crunching, which would muddy the pedagogical waters. I show the CB for our block of wood for several different orientations in figure 5.5. That the CB depends on heel angle, whereas the CG does not, has the following consequence. When the two centers are not aligned vertically, the force of gravity and the force of buoyancy (which have the same magni-

Figure 5.6. (a) Center of gravity, CG (open circle), and center of buoyancy, CB (x), for a floating box. If the box is a solid block of wood, the CG is at the geometrical center, in this case chosen to be at the water line. Arrows indicate upward buoyancy force and downward gravity force. These forces are not aligned and therefore apply a torque to the box. The horizontal separation of CG and CB is the righting arm, denoted GZ. (b) If the box was originally in a stable position (outlined in gray), a line from the CG to the top would appear on the rotated box as shown by the slanted dotted line. This line intersects the "new" vertical from the CB at the metacenter (triangle). The distance between CG and the metacenter is metacentric height, denoted GM. If the metacenter is higher than the CG, as here, the box orientation is stable—it will return to its original position. (The righting arm, GZ, is defined as positive or negative depending on whether the metacenter is above or below the CG.)

Figure 5.6. (a) Center of gravity, CG (open circle), and center of buoyancy, CB (x), for a floating box. If the box is a solid block of wood, the CG is at the geometrical center, in this case chosen to be at the water line. Arrows indicate upward buoyancy force and downward gravity force. These forces are not aligned and therefore apply a torque to the box. The horizontal separation of CG and CB is the righting arm, denoted GZ. (b) If the box was originally in a stable position (outlined in gray), a line from the CG to the top would appear on the rotated box as shown by the slanted dotted line. This line intersects the "new" vertical from the CB at the metacenter (triangle). The distance between CG and the metacenter is metacentric height, denoted GM. If the metacenter is higher than the CG, as here, the box orientation is stable—it will return to its original position. (The righting arm, GZ, is defined as positive or negative depending on whether the metacenter is above or below the CG.)

tude but opposite direction) do not quite cancel out but instead exert a torque on the hull. This torque also depends on the heel angle, which complicates the physical analysis significantly.

We can quantify this dependence of the buoyancy force on the heeling angle in a manner that readily conveys information to the eye, and so is very popular among boat designers. Much of a boat's heeling characteristics can be captured in a graph called a righting arm curve. The righting arm is the horizontal distance between CG and CB, as shown in figure 5.6. It is conventional to denote righting arm by the abbreviation GZ. This distance is important because it is proportional to torque:

increased GZ means increased torque. If GZ is positive then the torque is a righting moment that acts to reduce a boat's heeling angle; if negative then the torque will act to capsize the boat.

How do we decide whether GZ is positive or negative? In other words, how do we know if the torque that is applied by the combined forces of buoyancy and gravity will act to right a boat or capsize it? The answer lies in yet another technical term: metacenter. The metacenter is explained in figure 5.6. The concept is slightly abstract, so I will provide a separate definition of it here. Consider Puddleduck sitting calmly on a flat sea, with her mast vertical. A large wave comes along that causes her to heel over a large amount, as happened to the block shown in figure 5.6. Puddleduck's center of buoyancy moves because when she is heeling, her hull displaces water in a different way than when she is upright, as we have seen. We find her metacenter at the new heeling angle by a simple geometrical construction. The CB acts vertically: we extend a vertical line from the CB to a line through her mast. (Of course, now that she is heeling, her mast is no longer vertical.) The point of intersection of these two lines is the metacenter, denoted by a triangle in figure 5.6.

Here is the significance of the metacenter: if it lies above the center of gravity, the torque will right Puddleduck: she will return to her stable upright position. If the metacenter lies below the CG, the torque will cause her to capsize. Of course, the location of the metacenter changes with the heeling angle because the CB changes; that makes it difficult to say what is going to happen by quickly glancing at the equations. However, the stability information is readily conveyed in Puddleduck's righting arm curve, which is a plot of GZ vs. heeling angle. Determining the righting arm curve for a complicated structure like a boat is difficult and depends on every last detail of hull shape and mass distribution. So instead, in figure 5.7, I show the righting arm curve for our block of wood. You can see that the shape of the curve changes as the shape of the block changes. This change is observed also in boats: wider boats show more initial stability, meaning that they are more difficult to tip over. The righting arm curve displays this initial stability in the steepness of the slope near zero heeling angle, amz = 0°. A more stable block or boat has a steeper slope because it has a greater righting moment and so is harder to tip over. Catamarans are very wide and have great initial

40 60 80

Heeling angle (degrees)

40 60 80

Heeling angle (degrees)

Figure 5.7. (a) Righting arm length, GZ, vs. heeling angle, amz, for different floating boxes. If the center of gravity of the box is lowered from the geometrical center (appropriate for a solid block) halfway to the bottom (more appropriate for a boat), the dashed curves result. Black lines correspond to box width and height of (w,d) = (3,2), and gray lines to (w,d) = (3,1). (b) Movement of CB from (x,z) = (0,0) as the heeling angle increases from 0° to 120°.

Figure 5.7. (a) Righting arm length, GZ, vs. heeling angle, amz, for different floating boxes. If the center of gravity of the box is lowered from the geometrical center (appropriate for a solid block) halfway to the bottom (more appropriate for a boat), the dashed curves result. Black lines correspond to box width and height of (w,d) = (3,2), and gray lines to (w,d) = (3,1). (b) Movement of CB from (x,z) = (0,0) as the heeling angle increases from 0° to 120°.

stability. More slender boats with deeper hulls, like our block of wood in figure 5.6 (with its righting arm curve also plotted in figure 5.7), have less initial stability. They tip over more easily and, as we will soon see, roll more; their roll amplitude and roll period (the time taken to roll back and forth once) are both greater than for a flatter boat, or block.

Note from figure 5.7 that the righting arm curve peaks at a certain heeling angle, corresponding to maximum righting moment, and then decreases as the heeling angle increases further. At these larger angles the boat or block will still try to right itself, so long as the righting arm curve is positive, but the torque is reduced. At some larger heeling angle the curve reduces to zero. At this point there is no righting moment, and the boat or block will stay at this angle if placed there: there is no torque to cause it to turn either way. At still larger heeling angles the curve is negative, corresponding to an overturning or capsizing torque; when a boat or block gets into this region, it cannot right itself. The heel angle at which the righting arm curve dips to zero marks the range of stability for our boat or block. The two characteristics that emerge from righting arm curves—initial stability and range of stability—are key design parameters for boat-builders. Obviously, we would like both to be as large as possible, but it doesn't work like that. I have tried to indicate in figure 5.7 how boat curves differ from those of wooden blocks by lowering the CG of the blocks. This is what would occur if the blocks were hollowed out to form a crude boat hull. The results show as better righting arm curves: greater initial stability, increased righting arms and hence increased righting torque, and greater range of stability. From figure 5.7 we see that flatter, hollow blocks have greater initial stability but a lesser range of stability, and the same is generally true of boats. The message is: go to sea in a boat, not a block—but I guess you already knew that.

Boats differ from blocks in other ways too. The downflood angle is the maximum heeling angle that a boat can have before she swamps. For an open boat, this angle can be quite small: the righting arm curve abruptly ceases instead of smoothly varying out to large heeling angles, as in figure 5.7. The watertight deck on many modern yachts permits much larger heeling angles. Even here, of course, it is necessary to close the hatches. Hatches tend to be placed along the centerline of a yacht, and as high up as practicable, to permit large heeling angles without danger of swamping. Another difference between boats and blocks is much more difficult to quantify. Boats carry movable masses which can shift as the boat heels. Whether it be gas or water in tanks, or cars and trucks on ferries, shifting mass can be very dangerous in a rolling boat. What seems stable can become unstable very quickly, and in the past a number of tragedies have resulted for this reason. To avoid the stability problems that can result from liquid sloshing back and forth, fuel tanks on yachts have baffles placed in them.

Was this article helpful?

Lets start by identifying what exactly certain boats are. Sometimes the terminology can get lost on beginners, so well look at some of the most common boats and what theyre called. These boats are exactly what the name implies. They are meant to be used for fishing. Most fishing boats are powered by outboard motors, and many also have a trolling motor mounted on the bow. Bass boats can be made of aluminium or fibreglass.

## Post a comment