The Geometry of Ships

THE GEOMETRY OF SHIPS 47
Define the nominal lengths of the three parent
bodies as:
L 1 X 1 X 0 , L 2 X 2 X 1 , L 3 X 3 X 2 ;
their waterline lengths as:
W 1 1 L 1 , W 2 L 2 , W 3 3 L 3 ;
their displacement volumes as:
V 1 , V 2 , V 3 ;
and their centers of buoyancy as:
1 L 1 , X 1 0.5L 2 , X 2 3 L 3 .
( 1 , 3 , 1 , and 3 are all constants that depend on the
original ship geometry.)
Now apply longitudinal affine stretching factors 1 ,
2 , 3 to the three bodies, and reassemble them into a
complete candidate ship. The result of this construction
is a triply infinite family of candidate ship forms, with 1 ,
2 , 3 as parameters. (The parent ship is 1 1, 2 1,
3 1.) To determine these parameters, we will impose
an equal number of conditions (form parameters):
• Displacement volume, T
• Longitudinal center of buoyancy (as a fraction of waterline),
T
• Prismatic coefficient, C pT .
(The subscript T stands for “target.”)
Next, we need a way to evaluate the form coefficients
as functions of 1 , 2 , 3 . The properties of affine transformation
make this easy. First, the waterline length W
is the sum of the body waterlines:
W 1 1 L 1 2 L 2 3 3 L 3 (114)
Displacement volume is the sum of the three body
volumes:
T 1 V 1 2 V 2 3 V 3 (115)
Likewise, the x-moment of displacement volume is
the sum of the body volumes, each multiplied by the X-
coordinates of its respective centroid:
M X 1 V 1 [ 1 1 L 1 ]
2 V 2 [ 1 L 1 0.5 2 L 2 ]
3 V 3 [ 1 L 1 2 L 2 3 3 L 3 ]
T T W (116)
The prismatic coefficient is:
C pT = T / [A ms W] (117)
where A ms is the midship section area.
Equations (115, 116, 117) are three simultaneous
equations in the three unknowns 1 , 2 , 3 . (Note that in
general, the equations are likely to be nonlinear, though
in this case all but equation (116) can be arranged in linear
form.)
Such a system of simultaneous nonlinear equations
can be attacked with the Newton-Raphson method
(Kreyszig 1979; Press, Flannery, Teukolsky & Vetterling
1988). With any luck, this will provide an efficient and accurate
solution. Some numerical pitfalls should be noted.
When the equations are nonlinear, there is no guarantee
that a solution exists; even when they are linear, there is
no guarantee of a unique solution. Convergence to a solution
can depend on the values used to start the iteration.
In this example, a solution with any of the ’s less
than zero would not be a meaningful result.
A form parameter-based system can also be built
around a general optimization algorithm (Kreyszig
1979; Press, Flannery, Teukolsky & Vetterling 1988),
which seeks to minimize some objective function such
as predicted resistance at a specified operating speed,
or an average surface fairness measure, with equality or
inequality constraints stated in terms of various form
parameters.
During the design of a vessel, the methods of hydrostatic
analysis detailed in Section 9 are applied to tabulate and
graph various hydrostatic properties. This information is
used throughout the design process to assess the hydrostatic
equilibrium and stability. If the vessel is subject to
classification, hydrostatic properties must be submitted
as part of that procedure. Further, hydrostatic properties
will be communicated to the owner/operator of the vessel
to be utilized during loading and operation. In the
eventuality of a collision or grounding, knowledge of hydrostatic
properties may be crucial in the conduct and
success of salvage operations. Because the data will be
Section 11
Upright Hydrostatic Analysis
used for several functions beyond the design office, it is
important that it be developed and furnished in a more or
less conventional and agreed-upon format. Such formats
are well established for conventional vessel types. In the
case of an unconventional vessel, it may be a challenge to
decide on a relevant set of hydrostatic properties, and to
present them in such a way that users of the information
can relate them to the standard conventions.
The input to the hydrostatic calculation is in most
cases a form of offsets on transverse stations, represented
in a computer file. It is important to document the
actual offsets used. Graphic views of the offsets are ben-