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A Calculation of Trim

On page 115 in the Guilmartin's Book "Galleons and Galleys", it is described that "-----; no small matter with a gun weighting 7,000 pounds mounted on the bow of a hull displacing 170 tons, figures representative of mid sixteenth-century Venetian triremes. Shipwrights compensated for the weight by designing hulls that were fuller at the bow and finer at the stern, giving the underwater lines a graceful , fish-like shape."
I am interested in this description. So, I try to calculate the trim of the galleys.

Therotical background1),2),3)


When all of the guns move from the center to the bow, the waterline rotates around the y-axis at the center of flotation "F" as shown in the figure.
trim


The metacenter M is the pivot point of the entire body for small angles of rotation around y-axis. The distance between the metacenter and the center of buoyancy B or B' is as follows.
eq1

where A is waterplane area, V is underwater volume
x=0 is at the center of flotation "F"

On the otherhand, the distance between the center of gravity G and G' is as follows.
eq2

where delta is displacement.

Since rotation angle is small, following equations are derived.
eq3,4

Therefore, trim t is described as follows.
eq5

Generally, we can assume
eq6

Therefore
eq7


The center of flotation "F" is the geometric center of the waterplane area. Therefore, xc is obtained by following equation.
eq8

where x=0 is at the stern

Results

The calculated results are summarized in next table.
I assumed that the displacement delta=170 tons and the weight of the guns w=3.2 tons (7,000 pounds) .

waterplane shape (y-axis is enlarged) BM t tf xc
60.3 m 18.7 cm 9.3 cm 0m
100.9 m 11.2 cm 5.2 cm 1.28 m
114.7 m 9.8 cm 4.6 cm 1.43 m
128.6 m 8.8 cm 4.4 cm 0 m
54.3 m 11.7 cm 5.9 cm 0 m
16.1 m 17.6 cm 8.8 cm 0 m
22.5 m 28.2 cm 14.1 cm 0 m
8.9 m 31.8 cm 15.9 cm 0 m

BM : Distance between the metacenter and the center of buoyancy
t : trim
tf : Bow sinking depth from the initial waterline plane
xc : Distance between the center of flotation "F" and the center of the body

Conclusion

Very simple shapes have been used in this calculation. So, absolute value is not so important.
But we can see a tendency that the shape with a fat bow and a slim stern reduces trim.
Since the freeboard of the galley was very narrow, the several centimeters improvement of the trim mght be very important.
Therefore, I think this calculation supports Professor Guilmartin's description.
Reference:
1) "Ship Hydrostatics, Equilibrium and Stabilty: Basics".
2)Correspondence study course of ship building (in Japanese)
3)2.9.3 Longitudinal Center of Floatation (LCF)