Page 31 - Introduction to Naval Architecture

3 Ship form calculations

It has been seen that the three dimensional hull form can be
represented by a series of curves which are the intersections of the hull
with three sets of mutually orthogonal planes. The naval architect is
interested in the areas and volumes enclosed by the curves and surfaces
so represented. To find the centroids of the areas and volumes it is
necessary to obtain their first moments about chosen axes. For some
calculations the moments of inertia of the areas are needed. This is
obtained from the second moment of the area, again about chosen
axes. These properties could be calculated mathematically, by integra-
tion, if the form could be expressed in mathematical terms. This is not
easy to do precisely and approximate methods of integration are
usually adopted, even when computers are employed. These methods
rely upon representing the actual hull curves by ones which are defined
by simple mathematical equations. In the simplest case a series of
straight lines are used.

APPROXIMATE INTEGRATION

One could draw the shape, the area of which is required, on squared
paper and count the squares included within it. If mounted on a
uniform card the figure could be balanced on a pin to obtain the
position of its centre of gravity. Such methods would be very tedious but
illustrate the principle of what is being attempted. To obtain an area it
is divided into a number of sections by a set of parallel lines. These lines
are usually equally spaced but not necessarily so.

TRAPEZOIDAL RULE

If the points at which the parallel lines intersect the area perimeter are
joined by straight lines, the area can be represented approximately by
the summation of the set of trapezia so formed. The generalized
19