Reropdacionde Albryferia;

In 2.7 and 3.3 we used the Cp reverse formula to check, with rough approximation, whether the immersed volume of the hull matches a realistic displacement for the type of yacht that we’re designing. It’s time now to calculate exactly the hull displacement. I’m afraid I’ve to say it again: I know that all the dedicated design programs on every computer will supply the designer with the data about volumes, displacements, stability and many more elements for several different immersion values, both for a straight hull and for an inclined one. Yet I feel that understanding the process through which it happens is fundamental, and it might also help to compensate for the contingent lack of the beloved computer. Imagine that the Aliens have taken over our Earth and that no more computer is working: we still have some elementary tools, such as paper, pencil, fingers for counting, the multiplication table by heart and our brains.

Please note that figure # 15 shows that part of the transverse sections lies below the LWL (it’s the quick works), while another part is above it (the dead works). Let’s forget about the latter: our target is to establish what immersed volume results from the integration of the transverse sections below the LWL. We use the Simpson’s rule: we’re talking of Mr. Thomas Simpson, not of Homer, of course.

This rule for the integral calculus is simple enough and was named after the 1700 British mathematician, but the Italian Bonaventura Cavalieri and the German Johannes Kepler competed for its authorship.

Fig. 15

It’s one of the many possible rounded-up ways to calculate the volume of an irregular-shaped body. All systems (Simpson’s or the trapezoidal rule) are based on the averages law. First of all we divide the LWL in an even number of spaces, which gives an uneven number of stations. Let’s have a look at the hull in 3.5: the common interval is 0.84 metres. As for the Simpson’s rule we should set the last station in correspondence to the LWL forward end: but there’s a bow bulb, which sticks out and needs to be considered while calculating the immersed volumes. Therefore we exceptionally set the last station in correspondence to the bulb edge. We get 28 spaces and 29 stations, numbered from

# 0 to # 28 (see figure # 15).

It’s time now to draw the Simpson’s chart table, as per figure # 16: and yes, I must acknowledge that a computer spreadsheet would be helpful (provided the Aliens allow us to use it).

The first column on the left, which title is “Stations”, lists the numbers of the transverse sections, or stations.

The second column’s headline is “1/2 areas” and it is void, for the time being.

The third column is assigned to “SM”, which stands for “Simpson’s Multipliers”. The rule is simple but binding: the first and last numbers must be 1 and all the other shall come in succession: 4, 2, 4, 2, 4 …

Column # 4 title is “SF”, which stands for “Simpson’s Functions”: it’s empty now, but we shall fill it with the product obtained multiplying each half area by the corresponding Simpson’s Multiplier.

The fifth column is assigned to “M”, meaning “Multiplier”. This time the numbers are the same as the stations.

Column six is entitled “Sm” (please note that “m” is now a lower case letter) and stands for

“Simpson’s moments”: it shall later list a number of “moments” (I wish to recall that a “moment” is the product of a force by a distance, or “arm”).

The top right corner reminds that the “CI” (aka “Common Interval”) is 0.84 metres.

The cells that shall brief the figures of volume, Mediterranean displacement, Gulf displacement and LCB are already set at the left bottom corner of the chart table. We must now fill the second column with the half areas of the immersed sections, corresponding to each station.

Yes, I know that nowadays any computer program will measure these areas quickly and with extreme accuracy, but before the birth of these processors, designers used a “planimeter” (see figure

# 17).

Fig. 16

The draftsman had to follow the contour of a two-dimensional figure, centring the line with a small red dot within a magnifying lens. Then the reading was to be multiplied by a correction factor for the scale and the result was the area of the figure. The values were plotted on a chart, to check that there were no blatant misreadings (see figure # 18). In my office there’s still planimeter, hidden in a drawer: you never know (the Aliens could be behind the corner). Nowadays reading the half areas figures by means of a computer dedicated program is a lot easier and quicker, of course. In any case I suggest that you double check, following

Principle number eight: checking twice never killed anybody.

Or, if you prefer, confidence is good, checking is better.

Fig. 17

We insert the half areas values, either read from the curve or from the computer, in the second column of the chart table (see figure # 19).

Fig. 18

Now let’s multiply the half areas figure by the corresponding Simpson’s Multiplier and write the result in the Simpson’s Functions column: for example (let’s take an easy one) the figure corresponding to station # 7 shall be 2.008 * 4 = 8.032 . The chart table looks now like in figure # 19.

Fig. 19

We sum all the Simpson’s Functions, getting a 174.357 figure. We are now able to calculate the immersed volume of the hull: the formula is (∑SF * CI *2)/3.

Fig. 20

We need to multiply by 2 because we have only read half the areas: dividing by 3 averages the

Simpson’s Multipliers. The actual formula becomes (174.357 * 0.84 * 2)/3 = 97.640 m³ which, multiplied by the specific weight of sea water gives for a 97.640 * 1.023 = 99,885 kg displacement.

I wish to highlight the importance of keeping the units of measure under control: if it’s metres, let it always be metres and let kilograms always be kilograms. Don’t change horse halfway the race.

Fig. 21

Let’s finally check where is the longitudinal centre of buoyancy, aka LCB.

We multiply the Simpson’s Functions (SF) by the corresponding Simpson’s multipliers (Sm), which determine the distance of each value from a common origin, aka station # 0. See figure # 21.

The formula is ∑Sm * CI/∑SF or, in other words, we multiply the sum of the Simpson’s moments by the Common Interval and then divide the result by the sum of the Simpson’s Functions. In the case in point we have 2453.558 * 0.84 / 174.357 = 11.82, meaning that for the designed waterline the LCB position is 11.82 metres ahead of station # 0. It goes without saying that the LCB positions changes considerably once we consider different immersions of the hull. It’s actually necessary to re-do the complete procedure for waterlines set at different levels and plot the results on a curve. Let’s then hope that Alien never bothers our computer, so that it does the dirty job while we play 18 golf holes!

We now know the longitudinal position of the Centre of Buoyancy: we need its vertical position too (aka VCG), to ascertain the vessel’s stability. It’s not difficult: just boring. We must repeat the same procedure using the waterlines areas instead of the stations areas.