The Geometry of Ships

50 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
An exception to positive initial stability occurs sometimes
in unconventional high-speed craft which are
partially or completely supported by dynamic lift at operating
speed. The geometric requirements on form for
high-speed operation may dictate a shape with negative
initial stability; such a vessel is said to “loll” to one side
or the other when floating at rest.
Sailing vessels require substantial roll stability to
counter the steady heeling moments associated with
wind force on their sails, and the opposing hydrodynamic
forces on hull and appendages. Roll stability is
therefore a crucial factor in sailing performance, and
many features of sailing yacht design have the purpose
of enhancing it: shallow, beamy hull forms, deep draft,
concentration of weight in deeply placed ballast, weight
savings in hull, deck, and rigging, and multihull configurations.
It is not initial stability that counts, of course,
but rather righting moment available at operating heel
angles; however, for conventional monohull designs, the
initial stability is a good indicator of sail-carrying power.
Some of the highest-performance sailing craft, trimarans,
can have negative initial stability.
11.1.7 Longitudinal Metacenter. Analysis of longitudinal
stability is highly analogous to that of transverse
stability. From equation (123),
dM / d pg (z M l z G ) GM L (124)
where z M l z B I yy /, the longitudinal height of metacenter,
and GM L is the longitudinal metacentric height.
The geometric term I yy / is longitudinal metacentric
radius. Note that I yy is the moment of inertia of the waterplane
about a transverse axis through the center of flotation
C F . If moment of inertia I YY is calculated with respect
to some other transverse axis, the parallel-axis theorem
must be used to transform to the center of flotation:
I yy I YY A wp xF 2 (125)
Normally, longitudinal stability is not a large design
issue because of the elongated form of most vessels. It
can be used operationally to predict the effect of longitudinal
weight movements and loading on the trim angle.
This can be expressed as moment to change trim 1 cm:
MT1cm GM L /(100L) (126)
11.1.8 Wetted Surface. For a vessel floating on a
specified waterline, the total area of its outer surface in
contact with the water is known as its wetted surface. As
this is an area, its units are length squared. Wetted surface
is of interest for powering and speed prediction; the
frictional component of resistance is ordinarily assumed
to be in direct proportion to wetted surface area. It also
indicates the quantity of antifouling paint required to
coat the vessel up to this waterline. Wetted surface is
often included in the curves of form.
The calculation of area for a parametric surface has
been outlined in Section 4.2, in terms of the components
of the metric tensor. This calculation is fairly straightforward,
though a complication is that the wetted surface is
usually only a portion of the complete parametric hull
surface, so the domain of integration in u, v space will
have a complex boundary which has to be computed by
intersecting the surface with a plane. In relational geometry,
it is convenient to create a subsurface or trimmed
surface (portion of a surface bounded by snakes, one of
which can be an intersection snake along the waterline)
representing the wetted surface; this can be arranged to
update automatically as the draft is varied.
Traditionally, wetted surface is calculated as the sum
of area elements over the hull surface, a double integral
with integration over x done last. The integral is not in a
form that can be reduced by Gauss’ theorem, so the integrand
turns out to be relatively complex compared with
most of the integrals required for hydrostatics. On a symmetric
hull, we can locate any given point X on the hull
by two coordinates:
(a) x the usual longitudinal coordinate
(b) s arc length measured from the centerline along
the intersection of the hull with the transverse plane
through x.
In terms of these (dimensional) parameters, the surface
point is described as X {x, y(x, s), z(x, s)}, and the
first derivatives are X x {1, y x , z x } and X s {0, y s , z s },
where subscripts x and s stand for partial derivatives.
The metric tensor components (equation 33) are
g 11 1 y 2 x z 2 x (127)
g 12 y x y s z x z s (128)
g 22 y 2 s z 2 s 1 (129)
(the last because s is defined as arc length in the transverse
plane), so the metric tensor discriminant becomes:
g 1 (y x z s z x y s ) 2 (130)
This can be expressed in terms of the components of
the unit normal n {y x z s z x y s , z s , y s }/ g as: g 1
nxg 2 and solving for g: g 1/(1 nx) 2 sec 2 , where is
the bevel angle with respect to the x axis. Thus the wetted
surface area is
WS g dx ds sec ds dx (131)
The sec factor is termed obliquity. For a sufficiently
slender hull, n x is everywhere small and sec will not
differ appreciably from 1. Then the wetted surface is (approximately)
simply WS G(x)dx, where G(x) is wetted
girth at station x.
Wetted surface does not transform in any simple way
under affine stretching. Under a geosim transformation,
being an area, it varies simply as 2 .
Wetted surface is conventionally nondimensionalized
with a combination of length and displacement to make
the wetted surface coefficient:
C WS WS/L (132)
where is displacement volume and L is length. Values
of C WS range from about 2.6 to 2.9 for usual ships of normal
form at design flotation.