International Moth Transverse Stability |
The aim of this article is to answer the question; is transverse stability an important factor in Moth design, and to increase our understanding of the influence that various factors have over the dynamic stability of the craft. To understand the craft's stability fully we must therefore analyse all the components that contribute to its stability
The sailor has a mass of seventy kilograms at a lever to the centreline of approximately 1.425 metres, if the heel angle (H) is zero. This gives a moment of
70×9.81×1.425 = 979 Nm
This shall be the magnitude of one unit moment. In this case it is acting in a positive clockwise direction. The sailor is at an offset angle F so the moment will vary according to the relationship
cos(W+F)
A realistic value of F is thirty degrees so the sailor’s mass unit moment is equal to
cos(H+30)/cos(30)
The Wolston Unit’s hydrostatics program was used to evaluate the stability of the Aussie Axeman hull form heeled up to seventy degrees. Seventy degrees is assumed to be the point of capsize. To isolate the GZ data (hydrostatic hull form stability) the centre of gravity of the two designs was put on the waterline. The GZ righting moment is then corrected by adding the realistic centre of gravity and dividing by 979 Nm to make it into a unit form. This vertical centre of gravity of the rigged craft is assumed to be 75 cm above the waterline so it’s component will be equal to
30×9.81×0.75×sin(H)
There is a float on the ends of both wings; these will provide 0.4 units of moment when submerged. A realistic assumed value of the heel angle at which they submerge is 25 degrees.
2.25/2L = cos(25)
L = 1.24 m
0.4×979 = 1.24×B
B = 316 N = 32 kg
The dimensions of the floats are approximately 2 m by 0.16 m by 0.1 m. This gives a buoyancy of
2×0.1×0.16×1025 = 32.8 kg
Note that the immersion of the floats will reduce the displacement of the hull to 67 kilograms, this will decrease the magnitude of the hulls GZ curve. This effect has not been incorporated into the analysis because, as shall be seen, it is small compared to the magnitude of the other components of stability.
According to B. Deakin’s paper "The development of stability standards for UK sailing vessels" : "Navel Architects concerned with yacht performance assume that the heeling moment varies with cos(H). This assumption was validated in the wind tunnel tests for close hauled sailing conditions up to heel angles of about thirty degrees." So in unit form the moment for the sail and dagger board interaction for angles up to thirty degrees is equal to
-cos(H)
For angles greater than thirty degrees we shall also assume this relationship to be valid. Although this is not an accurate relationship in this region it will allow us to continue this crude analysis without extensive wind tunnel tests.
The moment from the sail acts in a negative anticlockwise direction and is equal to negative one when H is equal to zero. The total moment on the craft is zero when the craft is sailing close hauled and upright. This is to simulate the a quasi-static equilibrium sailing state. The total moment on the craft as it deviates slightly from this angle and the effect of the floats when angles are greater, can be investigated.
Calculated quasi-static stability.
| H | Unit sail moment | Sailor’s unit mass moment | Unit GZ moment | Unit wing buoyancy moment | Sailor + GZ + wing | Total moment |
| -30 | -0.866 | 1.155 | 0.078 | -0.400 | 0.833 | -0.033 |
| -20 | -0.940 | 1.137 | 0.055 | 0.000 | 1.192 | 0.252 |
| -10 | -0.985 | 1.085 | 0.029 | 0.000 | 1.114 | 0.129 |
| 0 | -1.000 | 1.000 | 0.000 | 0.000 | 1.000 | 0.000 |
| 10 | -0.985 | 0.885 | -0.029 | 0.000 | 0.856 | -0.129 |
| 20 | -0.940 | 0.742 | -0.055 | 0.000 | 0.687 | -0.253 |
| 30 | -0.866 | 0.577 | -0.078 | 0.400 | 0.899 | -0.033 |
| 40 | -0.766 | 0.395 | -0.096 | 0.394 | 0.693 | -0.073 |
| 50 | -0.643 | 0.201 | -0.100 | 0.376 | 0.477 | -0.166 |
| 60 | -0.500 | 0.000 | -0.093 | 0.346 | 0.253 | -0.247 |
| 70 | -0.342 | -0.201 | -0.091 | 0.306 | 0.014 | -0.328 |
The total moment curve has a negative gradient for small angles of heel, which indicates that any deviation away from upright will be amplified, and the craft is therefore unstable.
Heeling the mast to leeward will dramatically increased the stability of the craft. Although the analysis is not shown here the sailors righting moment curve becomes more in phase with the sail's heeling moment curve and as a result the gradient of the curve around the upright area is shallower. It has been shown that the craft becomes about two times more stable than the craft with an upright mast.
How does a boat that spends most or all of it’s time in a state of instability sail upright?
We all know that Moths are very demanding boats to sail. The main sheet and rudder must be adjusted constantly to keep the craft sailing at the right angle of heel. The sail and foils add a large component of damping into the system making the task of sailing possible.
An acceptable value of transverse stability must be defined. This value will vary between sailors and sailing conditions. It shall be concluded that in general the vertical centre of gravity height, wing bar angle, float size and heel angle of the mast are far more significant to transverse stability than GZ curves. The Axeman has sufficient stability in my opinion as well as most Moth sailors; so due to the overriding dynamics of the situation I shall conclude that transverce stability is not an important factor in Moth hull design.
Although this analysis has helped us to understand the dynamics of the situation it may be fair to say that it is fundamentally flawed. It has not attacked the real causes of capsizes, which occur during tacks, gibes or light winds. These happen because the sailor does not always have the damping effect of the sail working at its full potential. The additional stability of the floats and the large damping experienced when the wing is dragged through the water along with the sailor's skill are key components in these situations.
A Moth heeled to windward by 20° ( H= -20) will enable 19 % more righting moment to be achieved. This explains why in strong winds the boats are often sailed heeled to windward. The loss of efficiency of the rig and dagger board is played off against the greater righting moment. More righting moment means the rig can be sheeted in harder and a potential increase in driving force obtained.
In the limited wind speed range between being over and under powered there is a potential advantage in raking the mast to leeward by twenty degrees and heeling the boat to windward by twenty degrees. This advantage must be considered against the extra complication whist tacking and the additional weight of the extra rig control system. Tacking is hard enough without having to wave the mast through forty degrees. So as the range of conditions where this would be an advantage is small and, due to the practicality of operating the system effectively, I have concluded that the idea is not worth employing at this stage. However if there are any crazy buggers that wish to explore this option I for one will be very interested in their conclusions.
I hope this article has not been to long winded and that the maths has not been to boring, but that it has achieved it's aim and increased your understanding of our fantastic craft.