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. |