Upright centre of buoyancy
heel angle
Upright centre of buoyancy
Knowing B', the location of the point where the vertical through B' hits the centre plane M^ can be found, sec Fig 4.12. BM may then be measured from the figure and the remaining formulae for small angles applied.
Curve of static The curve of static stability represents the righting moment at varying stability angles of heel. An example of this is given in Fig 4.13. Since the moment differs from the lever arm only with respect to the constant Ag, the vertical scale could equally well represent GZ.
For small angles GM is constant and sin O ~ ® (in radians), so GZ is proportional to the heel angle, ie GZ = GM • sin O ~ GM ■ <3>. The slope of the GZ curve at the origin may thus be obtained by noting that the tangent should pass through the point GZ = GM for O = 1 radian, ie at 57.3°.
Another important aspect of the GZ curve is the maximum, which represents the largest possible righting moment of the hull. Obviously the yacht will capsize if the heeling moment exceeds this level.
Of great interest is the so-called stability range, which is the range of x angles for which a positive righting moment is developed. For larger angles the hull is stable upside-down.
It is also of interest to note that the area under the RM curve up to
B — Buoyancy force G = Gravity force
Max. righting arm
Max,
Heel angle [deg]
-Stability range-
Stable upside—do wn
Fig 4.13 The curve of static, stability- YD-40
Fig 4.14 CZ - Curves for Grimalkin and the Contessa 32
Heel angle [deg]
a certain angle represents the work, by waves for instance, needed to heel the hull to this angle.
Large differences are found in the stability curves for modern fin-kccl yachts and traditional V-shaped long keel ones. After the Fastnet Race disaster in 1979, a study was carried out at Southampton University, in which two yachts of similar size were compared. Both raced in Class V.
One was a cruiser-racer, the Contessa 32, while the other one was an extreme racer, Grimalkin, 30 foot LOA. _
Interestingly enough both yachts have the same GM = 0.85 m (as appears from Fig 4.14. which shows the GZ-curves). This does not mean, however, that RM is exactly the same for small angles, since the mass differs: 4600 kg for the Contessa 32 and 3800 kg for Grimalkin. At 1° of heel RM is 670 Nm (Newton-metres) and 550 Nm, respectively. It should be noted that the sail area is almost exactly the same for both "yachts.
A larger difference is found in the maximum GZ, which is about 40% higher for the Contessa 32. Converted into righting moment the difference is even larger. For the Contessa 32 RMmax occurs at about 80° and is equal to 30200 Nm, while for Grimalkin RMmax is only 17900 Nm at about 50°.
A more significant difference is also found in the stability range. The Contessa 32 is stable up to about 155° while zero righting moment occurs already at about 115° for Grimalkin. There is thus a very small range of angles, 25°, where the Contessa 32 is stable upside down, and the area between the RM curve and the horizontal axis is very small in this range. For Grimalkin the corresponding range is about 65° and the area is significant. This means that it is considerably more difficult to put the latter yacht into the upright position once it has capsized. The amount of work required by wind and waves is large, so this yacht may be expected to stay upside down for some time, perhaps a few minutes, while the Contessa 32 would return to the upright position almost immediately after a knockdown.
From this discussion it is clear that the traditional yacht is safer under rough conditions than the more modem one. In the following; paragraphs we will elaborate further on the effects of waves on stability, before we present some statistics and criteria for the seaworthiness of ocean-racing yachts.
Rolling A sailing yacht in a seaway moves in all six degrees of freedom, ie surge, sway, heave, roll, pitch and yaw. The first three are linear motions in the longitudinal, transverse and vertical directions, while the remaining three are rotations around a longitudinal, transverse and vertical axis, respectively. From a safety point of view, rolling is the most important motion, and it will be dealt with in this and the following section. More important for the added resistance in waves are the pitching and heaving motions, and these will be discussed in Chapter 5, in connection with hull design.
If a hull is given a heel angle in still water and is then suddenly released, the righting moment will immediately tend to put the hull upright. The hull starts rolling back to its upright position, but due to its inertia it will not stop when the heel angle is zero. Rather, it will continue to roll over to the other side, where an opposing righting moment develops. The hull then rolls back and forth, until the motion is damped out. In fact, for a sailing yacht, the damping is very large, so the motion dies rapidly.
This example contains many of the important features in connection with rolling excited by a seaway. Of great importance is the frequency with which the hull rolls in the still water test; the so-called natural frequency. The higher the stability, and the lower the inertia, the larger the natural frequency. It can easily be imagined that if the frequency of the waves hitting the hull in rough water is the same as the natural frequency (resonance), very large motions may result, at least if the damping is small.
This phenomenon is clearly borne out in Fig 4.15. The horizontal scale is the frequency of encounter of the waves divided by the natural frequency of the hull, and the vertical scale is the roll angle divided by the wave slope. Several curves are shown in the diagram, each one with a constant damping. Note that the lowest curves represent the largest damping.
Fig 4.15 Roll amplitude for varying frequencies and damping
If the frequency of encounter is low or the natural frequency high, small values are obtained on the horizontal axis. This is where all curves converge into a value of one on the vertical axis. The roll angle is then the same as the wave slope. This may happen for long ocean waves after a gale, where most hulls will follow the wave contour. A liferaft, with a very small inertia, ic high natural frequency, will follow the wave contour for much shorter waves of higher frequency also, since the value on the horizontal scale is still very low. At the other end of the spectrum all curves tend to zero. This is where the waves hit the hull at such a high frequency that it does not have the time to react, an unlikely situation for waves of any significant height.
A dangerous condition is when the frequency of encounter is close to the natural frequency, ie close to resonance. As appears from Fig 4.15
the roll angle may then be several times larger than the wave slope and the yacht may capsizc. We will now discuss the various means of avoiding this situation.
If the yacht approaches resonance, ie the frequency of encounter gets close to the natural frequency, one of these frequencies must be changed. The most straightforward way of doing this is to change the course. Since the frequency of encounter depends both, on the wave speed (and length) and the speed component of the yacht in the direction of wave propagation, changing the course will change this frequency. If the yacht beats to windward many more waves are met per minute than if it runs downwind with the waves. This technique of avoiding excessive roll is used also on large ships under severe conditions. Speed reductions are also possible, of course.
From a theoretical point of view the natural frequency may be changed by increasing or reducing either the stability or the inertia (or more precisely, the mass moment of inertia around a longitudinal axis). To avoid the resonance situation the natural frequency can be either increased or reduced. However, in conditions where the problem occurs it is better to move to the left in Fig 4.15, either by increasing stability or reducing inertia. If weights located at a high position are moved down to the bottom of the hull (which is probably closer to the centre of gravity) both these effects are accomplished.
The technique of avoiding resonance is closcly related to the operation of the yacht, while the other way to reduce roll, namely to increase damping, is the designer's task. Damping may be caused by three things:
• Friction between the water and the yacht.
• Generation of waves on the water surface.
• Generation of vortices from the keel, rudder, sharp bilges and sails.
This factor is by far the most important for sailing yachts.
Vortex generation depends partly on the shape of the sections (see Fig 4.16), but mainly on the size of the lateral area. Excessive rolling combined with low speed crcatcs large angles of attack of the flow approaching the keel and rudder, which then get overloaded and stall. These phenomena will be dealt with at some length in Chapter 6. For
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