Designer's Notebook:
DISPLACEMENT by Simpson's Rule
Continued...
Next the designer will estimate where he expects
the waterline to fall and calculate the area of the hull below the waterline.
If the boat is a simple rectangle the cubic content can be obtained by
multiplying the underwater height x width x length (in feet) to obtain the
volume in cubic feet. This volume is then multiplied by the weight of a cubic
foot of water to obtain the estimated displacement weight of the hull. If this
displacement weight does not match the weight that the designer previously
calculated, the waterline is is adjusted and the process repeated. Doesn't
sound that bad, but remember, a boat is not uniform in shape.
There are many ways to determine the volume of an
irregular shape; a computer with a proper program can solve the problem quickly
if the proper factors are entered. There are several other methods but
let's go back to a way of calculating volume that is basic and old as the
hills: Simpson's Rule.
Simpson's Rule is used to determine the square
area of the surface of a plane, such as the area of one cross section of a
hull. When the areas of a number of sections have been obtained, the cubic
contents of the whole can be calculated.
Let's use an example of how this rule is used
to determine volume, then displacement. Suppose a boat has a waterline
(estimated) length of 20'. This length is divided into an even number of
equally spaced parts drawn perpendicular to the waterline. Let's use 10
parts 2' apart and number them from left to right 0 through 10 (the
bow).
Next the areas in feet of these sections must be
determined. The designers use a tool called a "planimeter", with
dials and wheels that determines area when traced around an irregular plane.
But the area can also be figured by laying the drawing over graph paper to a
scale compatable with the drawing, counting the squares in the plane and
estimating partial ones. The area on one half of the boat section is determined
and multiplied by two. A typical boat will have no area at station #10.
After the areas of all of the 10 sections have been
obtained, list them in columnar form with the station number (#) in the first
column and three additional columns, labeled A, B, and C. List the area of each
station in column A. In column B enter the following numbers: Station 1# and
#10 enter a 1, all odd stations station a 2, and all even stations a 4. You
should have a column starting and ending with 1 and alternating 2 and 4. Kinda
sounds like voodoo doesn't it?
Next multiply each of the figures in the B column
by the area listed in column A. Put the result in column C. When all station
rows in column C have been figured, add them together. So far, the point of
this exercise has been to get the total of column C. The next step is to
multiply the column C total by the station spacing in feet, (2' in our
example), and the result divided by 3 to give the underwater volume in cubic
feet. Multiply this by the weight of water (62.5 or 64 lbs./cu. ft.) and the
result is the displacement in pounds.
That wasn't so bad was it? But, the
displacement figured does not equal the calculated weight of the boat. So
it's back to the beginning, adjust the waterline up or down, and try
again.
There are many ways to estimate the waterline with
reasonable accuracy and numerous attempts are seldom required. But our original
intent was to present or represent an old rule that figures displacement.
Simpson's Rule is a handy one and can be used to find the area of any
irregular plane as well as the volume.
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