Misleading, Bewildering and Unreliable Stability Criteria 11/97

Difficulties with Equilibrium Angle as a Measure of Stability

The ability to resist a heeling moment resulting from wind, passenger crowding, etc. is often measured by the heel angle which is induced by the heeling moment. However, requiring that this angle of equilibrium be less than some given angle such as 14 degrees or some characteristic angle such as margin line immersion, downflooding point immersion or half freeboard does not in itself guarantee finite stability with the heeling moment. It says nothing about what happens if the heel is increased a bit farther.
With some vessels in certain conditions it is possible to meet such a criterion and yet have virtually no stability beyond the angle of equilibrium. Therefore, if the criterion envisions an actual or possible operating condition, it must be augmented by some measure of stability with the heeling moment in effect. Possible augmenting provisions would be a minimum range of stability, a minimum area, or a mimumum residual righting arm.

Difficulties with Angle of Maximum Righting Arm

The angle at which a righting arm curve reaches its maximum value can never be known precisely. The reason for this is that it is a property of the derivative of the righting arm curve, not of the curve itself. A property such as the angle of equilibrium is an angle at which the righting arm curve is zero. The angle at maximum is the angle at which the derivative of the curve is zero.
While a mathematical curve may have a precise derivative, a curve representing data obtained by experiment always contains some degree of uncertainty. Differentiation always exaggerates the uncertainty just as integration diminishes the uncertainty.
Whether measured by inclining a physical ship or inclining a computerized model of the ship, the righting arm curve contains experimental errors which are exaggerated by differentiation. Therefore the angle of maximum righting arm cannot be known with a precision comparable to the precision of the angle of equilibrium.
The differentiated curve (which plots the slope of the original curve) is typically more "bumpy" than the original; and it may go through many more slope reversals than the original. The closer you look; i.e. the more samples or angles you take, the bumpier it gets. This is because for closely-spaced samples the predominant difference is the experimental error: an attempt to examine the "fine structure" of the curve finds only the "noise" of the experimental error. Indeed, if no distinction is made between the property being measured and the noise, the results will be unrepeatable and bewildering.
Take, for example, a righting arm curve which actually has a "ledge"; i.e. for some range of angles the arm stays constant. Adding to this the experimental error, you find that it is not quite constant; in practice there will most likely be an angle of maximum perceived value within that range. While it may be insignificant that this so-called angle of maximum is not quite true (in fact there is no angle of maximum in the range) it may be significant that a later attempt to find a maximum in the same range comes up with a different angle due to a different stream of experimental errors.
This fact causes difficulty when the angle of the maximum becomes the controlling limit when attempting to find a maximum VCG which just satisfies the limit. Each time a slightly different VCG is introduced, a different stream of experimental errors, exaggerated by the differentiation, influence the angle of the maximum. For some range of VCG variations, the experimental errors (which are not proportional to the change of VCG) will be more significant than the actual change in the curve do to the change of VCG. This tends to bewilder the process of solving for the maximum VCG since the relationship between the VCG and the limit margin has become "noisy" and when taken at fine enough intervals is essentially random.
It is unfortunate that the angle of maximum righting arm has been included in some stability criteria. While the computational difficulties are not unsurmountable, the results often bear the effects of the experimental error. This must be understood by those dealing with the results and appropriate allowances made.

GM at Equilibrium
When the equilibrium angle is not upright, increasing the VCG tends to increase the equilibrium angle. If it so happens that this new heel angle encounters new buoyancy of sufficient magnitude to greatly boost the transverse moment of waterplane inertia, the new GM may actually be higher than the original. Therefore, it is possible that the relationship between GM and VCG at equilibrium is the reverse of what would normally be expected.

Jumps in the Area to RA0
If you have a double-humped righting arm curve, and your area or area ratio limit puts the valley between the humps right at the axis (or just touching the heeling arm curve), there will be a discontinuity in the area and probably in the area ratio as the VCG is varied slightly. If the value called for in hte limit is in the discontinuity there will be no solution. The VCG which puts the valley just slightly positive may be considered acceptable; however, with a slightly higher VCG (or stronger wind) the vessel would hover at the angle of the valley -- not capsizing further but not returning to the initial equilibrium either.

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