The Geometry of Ships

THE GEOMETRY OF SHIPS 45
where:
L and M are the restoring moments about the longitudinal
and transverse axes respectively;
and are heel and trim angles;
I xx and I yy are the moments of inertia of the waterplane
area about longitudinal and transverse axes through
CF;
is the displacement volume;
z B and z G are the vertical heights of the center of buoyancy
and center of gravity respectively.
Because these coefficients pertain to small displacements
from an equilibrium floating attitude, they are
called transverse and longitudinal initial stabilities.
Their dimensions are moment/radian (i.e., force
length / radian). They are usually expressed in units of
moment per degree.
Initial stability is increased by increased moment of
inertia of the waterplane, increased displacement, a
higher center of buoyancy, and a lower center of gravity.
Because of the elongated form of a typical ship, the longitudinal
initial stability is ordinarily many times greater
than the transverse initial stability.
It is common to break these formulas in two, stating
initial stabilities in terms of the heights of fictitious
points called transverse and longitudinal metacenters M t
and M l above the center of gravity G:
dL/d g(z Mt z G ) (z Mt z G ) (108)
dM/d g(z Ml z G ) (z Ml z G ) (109)
where
z Mt z B I xx / (110)
z Ml z B I yy / (111)
z Mt z G and z Ml z G are called transverse and longitudinal
metacentric heights. There is an alternative
conventional notation for these stability-related vertical
distances:
B represents the center of buoyancy;
M T the transverse metacenter;
M L the longitudinal metacenter;
G the center of mass; and
K the “keel” or baseline.
Then,
BM T z Mt z B transverse metacentric radius
BM L z Ml z B longitudinal metacentric radius
KM T z Mt z K height of transverse metacenter
KM L z Ml z k height of longitudinal metacenter
KB z B z k center of buoyancy above baseline
KG z G z k center of gravity above baseline
GM T z Mt z G transverse metacentric height
GM L z Ml z G longitudinal metacentric height.
In terms of metacentric heights, in this notation, the
initial stabilities become simply:
dL/d GM T (112)
dM/d GM L (113)
Note that the metacenters are widely different for
transverse and longitudinal inclinations. Metacentric
heights are typically 10 to 100 times larger for longitudinal
inclination, owing to the elongated form of most
vessels.
It is conventional in naval architecture to compare vessels
of different sizes and proportions in terms of a number
of ratios or dimensionless coefficients characterizing
the form or shape. These so-called form coefficients correlate
to a useful degree with resistance, seakeeping and
capacity characteristics, and provide considerable guidance
in selecting appropriate proportions and displacement
for a new ship design.
The leading dimensions involved in the standard form
coefficients are: displaced volume , waterplane area
A wp , midship section area A ms , length L, waterline beam
B, and draft T. Any of these quantities might be very
clearly defined or might be ambiguous to varying degrees,
depending on the type of vessel and its specific
shape; these issues were discussed in Section 1.2.1 in relation
to “particulars.” For example, appendages might
or might not be included in displacement and/or length.
A ms may refer to the midship section (at the midpoint of
L) or to the maximum section, which can be somewhat
Section 10
Form Coefficients for Vessels
different. Of course, any uncertainty in the leading dimensions
will produce corresponding variations in their
ratios. To be definite about form coefficients, it is necessary
to explicitly state the loading condition and, often,
to specify which volumes are included and excluded. It
is common to refer to “bare-hull” or “canoe-body” form
coefficients when appendages are excluded.
10.1 Affine Stretching. A given ship form can be
transformed into a triply infinite family of other ships by
a combination of linear (uniform) stretchings along the
three principal axes. Uniform stretching by different
amounts along different axes is called affine transformation
in geometry. Suppose we start with a base ship form
of length L, beam B, and depth D and apply multiplicative
factors of , , along the longitudinal, transverse,
and vertical axes respectively; then we arrive at a new
ship with leading dimensions L, B, D. The displacement
will be multiplied by a factor of , the midship
section area by a factor of , and the waterplane area