The Geometry of Ships

40 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES
9.3 Planimeters and Mechanical Integration. During
the centuries in which graphical design operations were
so central to ship design, an important traditional tool of
the naval architect has been an area-measuring mechanical
instrument known as a planimeter. This is a clever
device with a stylus and indicator wheel; when the user
traces one full circuit of a plane figure with the stylus, returning
to the starting point, the indicator wheel rotates
through an angle proportional to the area enclosed by
the figure.
More complex versions of this instrument, known as
integrators, are able to additionally accumulate readouts
proportional to the moments of area and moments
of inertia of the figure. The previous edition of this book
contains a mathematical derivation of how the planimeter
works. Today, with the great majority of area, volume,
weight, and hydrostatic calculations performed by
computer programs, planimeters are likely relegated to
the same dusty drawer as the slide rule.
9.4 Areas, Volumes, Moments, Centroids, and Moments
of Inertia. Volume is usually calculated as an integral of
areas. In the general volume integral
V
dxdydz,
(57)
Fig. 33 Numerical integration rules. (a) Sum of trapezoids. (b)
Trapezoidal rule. (c) Simpson’s first rule (being applied to rather unsuitable
data).
constant, x i x i1 x, and the sum of trapezoids
takes the simpler form (the “trapezoidal rule”):
x0
(55)
... 2y N1
y N
)
Note: The trapezoidal rule can be seriously in error if
the function has discontinuities; in such cases, the sum of
trapezoids will usually give a much more accurate result.
9.2.3 Simpson’s First Rule. When (1) the tabulation
is at uniformly spaced abscissae, (2) the number of intervals
is even (number of abscissae is odd), and (3) the
function is known to be free of discontinuities in both
value and slope, then a piecewise parabolic function can
be a more accurate interpolant. This leads to “Simpson’s
first rule”:
xN
xN
x0
ydxx /2(y 0
2y 1
2y 2
ydxx /3(y 0
4y 1
2y 2
4y 3
... 2y N2
4y N1
y N
)
(56)
Note: When the three conditions above are not met,
Simpson’s rule can be much less accurate than the trapezoidal
rule or sum of trapezoids.
the integration can be performed in an y order. The
usual choice in ship design is to take the x axis longitudinal,
and integrate last with respect to x:
where
(58)
(59)
i.e., S(x) is the area of a plane section normal to the x-
axis at location x, the so-called section area curve or
section area distribution of the ship.
The area of an arbitrary plane region R in the x, y-
plane, enclosed (in the counterclockwise sense) by a
closed curve C R, is:
(60)
Green’s theorem allows some area integrals to be expressed
as line integrals around the boundary R. In general
2-D form, Green’s theorem is (Kreyszig 1979)
(Q / xP / y)/dx dy (P dx Qdy)
(61)
R
R
where P and Q are arbitrary differentiable functions of x
and y. One way to cast equation (60) into this form is to
choose P y and Q 0; then
A
VS(x) dx
S(x)
A dxdy
R
R
dydz
dx dy
y dx
R
(62)