3 Longitudinal Stability
A big problem with narrow Moth designs is that they tend to nose dive in strong winds. whilst sailing down wind. This is obviously to be avoided as the boat will tend to trip over its nose violently.
Putting more buoyancy at the front end of the boat is not really the answer as it will result in a fuller bow. A full bow will produce more drag when immersed. This will decelerate the boat and may make matters worse.
The best solution to the problem, to the authors knowledge, is to attach foils to the tip of the rudder. These should be designed to reduce tip losses on the rudder, produce lift when in a bow up planning state and exert a downwards force on the back of the boat when the nose dips down.
We shall examine the Axeman with no foils, sailing in a bow down state, to provide an acceptable standard of longitudinal stability.
3.1 Investigation into Longitudinal Stability
Assume the LCG of sailor at x = -1.5m
Assume LCG for boat to be x = 0
Schematic diagram of moment components considered in this analysis
The hydrostatic data in tables 3.1 and 3.2 was calculated using the Wolson Units Hydrostatics Programme
Trim = -0.2 m
|Table 3.1||WSH LCB position.||Table 3.2||Axeman LCB position.|
|Draft||D (kg)||LCB||Draft||D (kg)||LCB|
Axeman without foils
D = 100 kg
trim = -0.2m
Moments about LCB
[70 ´ (0.363 + 1.5) + 30 ´ 0.363] ´ 9.81 = 1386 Nm
WSH without foils
D = 100 kg
trim = -0.2m
moment about LCB
[70 ´ (0.178 + 1.5) + 30 ´ 0.178] ´ 9.81 = 1205 Nm
The difference between the WSH design without foils and the Axeman is -181 Nm
WSH with foils exerting 10 kg at x = -1.978
D = 110 kg
trim = -0.2m
moments about LCB
[70 ´ (0.168 + 1.5) + 30 ´ 0.168 + 10 ´ (1.978 + 0.168)] ´ 9.81 = 1405 Nm
The difference between the WSH design with foils and the Axeman is +19 Nm
3.2 Stabilising foil design
To maintain similar stability to the Axeman, the WSH design must have foils that exert 10 kg of force when the trim is -0.2 m and a realistic speed is under way.
The assumptions made about the foils operating conditions are as follows.
negative lift = 10 kg = 98.1 N
trim of boat = -0.2m
Draft (at x = 0) = 0.01m
Therefore the angle of attach of the foil is
tan a = 0.19 / 1.6775
a = 6.5°
The slowest speed of boat where nose diving may be a problem is estimated to be 6 knots. The lift generated by the foil can be calculated as follows.
L = ½ r s V² CL
L = lift = 10 ´ 9.81 = 98.1 Newton's
V = 6 knots = 3.1 m/s
r = 1025
CL = coefficient of lift
S = surface area of foil
98.1 = ½ . 1025 . (3.1)² S CL
S = 0.01992 / CL
CL is found from foil section data. We must chose a foil section that is suitable for this application. The section must operate without stalling between angles of ± 9° although it will mainly operate between ± 4°. The section must have high lift to angle of attack ratio. A minimum coefficient of drag for small angles is important so that this additional appendage does not slow the boat significantly. A symmetrical foil is desirable so that it will provide considerable lift when the boat has a bow up trim as well as negative lift when nose diving.
Using "The Theory of Wing Sections" by Abbot and Von Doenhoff, section data for various foils operating at a Renoulds number (Rn) of 3.0´ 106 was compared. (This is the closest Rn to the normal operating condition of the foil, see calculation below). NACA 65.012 to be most suitable section, its characteristics are shown in figure 3.1.
Rn = V L / n
V = realistic average speed = 5 knots = 2.57 m/s
L = cord length = 0.1m
n = kinematics viscosity = 1.194´ 10-6
Rn = 2.57 ´ 0.1 / 1.194´ 10-6 = 2.2´ 105
The following formula was taken from "Aero-Hydrodynamics of sailing" by C.A. Marchaj. It is true for untwisted foils with an elliptical profile.
CL = k a / [ 1+ ( 2 / AR )]
k = dL / da = 0.108 for NACA 65.012
AR = aspect ratio = 5
a = 6.5
S = 0.01992 ´ [ 1 + ( 2/5 )] / (0.108 ´ 6.5 )
S = 0.04 m2
A foil of surface area 0.04 m² is achieved by an elliptical foil of 11 cm cord and 45 cm span, which gives an aspect ratio of 5.
Area of ellipse = a b p / 2
AR = a2 / S
a = span of the foil = 0.45 m
b = root cord = 0.11 m
3.3 Longitudinal Stability Conclusions
Two rudders should be made for the boat, one with and one without tip foils. In light winds, where nose diving is not a problem, the tip foils could be dispensed with so that their unnecessary drag will not slow the boat.
The force provided by the foil is very speed dependant. The faster the boat travels the more effective it will be. Clearly the size of the foils will determine the level of stability of the craft; so any reasonable level can be obtained to the requirements of the sailor and the conditions.
Although the WSH design is not as stable as the Aussie Axeman an acceptable level of stability can be obtained with transverse foils on the rudder tip.