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lane, and consequently at the B. M. without the labor of the oregoing calculation by multiplying the Length by the Breadth by coefficient, which coefficient will be determined by a and selected from the table given on page 48. By referring to this able, we find for a (value .694) that the coefficient "i" (inertia coefficient) is equal to .0414, whence we get I= L × B3 × i = 100 < 123 x .0414= 7154 moment of inertia, which is sufficiently close for all purposes, and :—

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By transposing and taking the calculated I, we find

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Longitudinal Metacentre (L.M.C.)

From the definition given for the transverse metacentre it will be seen that if the ship be inclined longitudinally, instead of, as in the former case, transversely, through a small angle that the point in which the vertical through the altered C. B. intersects the original one will also give a metacentre known as the longitudinal, or L.M.C. Its principal use and value are in the determination of the moment to alter trim and the pitching qualities of the vessel, or longitudinal stability. It will be obvious that the moment of inertia of the water plane must be taken through an axis at right angles to the previous case, viz., at right angles to the centre line through the centre of gravity of water plane, which will be where the original and new water planes cross one another in a longitudinal view.

L.M.C. above C.B. =

I

of Water Plane about its C.G.
Volume of Displacement

Therefore, to calculate the M I1, we must figure the moment of inertia with, say, ordinate 5 (or any other one) as an axis when the moment about a parallel axis through the centre of gravity plus the product of the area of water plane multiplied by the square of the distance between the two axes will equal the moment about ordinate 5.

The moment of inertia about the midship ordinate we shall call I, and the distance of the centre of gravity from this station =x. The moment of inertia about the centre of gravity of plane = I1. We then have I=I1+ Ax2, or I1=I-Ax2. A clearer conception of this will be obtained from the tabulated arrangement.

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Area of water plane = 62.41 × (2 × 10) × 2.

=832.14 square feet.

Distance of centre of flotation abaft ordinate 5

=

(67.5752.41) 10
62.41

=2.42 feet.

Moment of inertia of water plane about ordinate 5

= 324.13 × (3 × 10) × 102 × 2 = 432,172 = I.

Moment of inertia of water plane about axis through its centre of flotation.

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An excellent approximate formula for the longitudinal B.M. is given by J. A. Normand in the 1882 transactions of the I.N.A. Taking the symbols we have been using :

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Applying this formula to the vessel with which we are dealing, we find :

L.B.M.=.0735

832.142 × 100
12 x 2583.7

164.12 feet.

which is a very close approximation to the calculated result of 165 feet.

We may also use the approximate formula which we applied in the case of the transverse B. M. altered to suit the new axis with a modified coefficient, as :

L.B.M. L3 × B× i1.

Moment to Change Trim (M1).

As the centre of gravity of the displacement (or centre of buoyoncy), either in the vertical or the longitudinal direction may be an entirely different locus from the ship's centre of gravity, it is obvious that unless the moment of the weights of the ship and engines, with all equipment weights, balances about the centre of buoyancy we shall have a preponderating moment deflecting the head or stern, as the moment is forward or aft of the C.B., respectively, until the vessel shall have reached a trim in which the pivotal point or C. B. is in the same vertical line as the completed ship's centre of gravity. To determine the moment necessary to produce a change of trim (M1) in a given ship, it is necessary to know the vertical position of the centre of gravity of the vessel and the height of the longitudinal metacentre (L.M.C.). The former may be calculated in detail or preferably proportioned from a similar type ship whose centre of gravity has been found by experiment; although great accuracy in the location of this centre in calculating the moment is not as important as in the case of G.M. for initial stability, as small variations in its position can only affect the final result infinitesimally. To investigate the moment affecting the trim, let us move a weight P already on board of the 100foot steamer whose calculations are being figured.

D= Weight of ship including weight P = 73.82 tons.
BM 165 feet.
P 5 Tons.
GM160 feet.

150 feet (distance moved).

L= 100 feet (length of vessel).

In the figure we have the centre of gravity G to G1, and the centre of buoyancy from B to B1, due to the shifting of the weight P forward for a distance represented by l, giving a moment

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The new water line is at W1L1 and B1G1 are in the same vertical and at right angles to it, and the point of intersection of the original and new water line at "O" equal to the centre of gravity (flotation) of water plane, therefore the triangles GMG1, WOW1 and LOL1, are of equal angle, so that

1

GG1 WW1 LL1 WW1 + LL .

=

=

=

GM WO LO WO+LO

But WW1+ LL1 is the change of trim, and WO+LO is the length of the vessel = L, then

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WW1+ LL1;
WO+LO

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Calling this change of trim 24 inches, and assuming that the point of intersection "O" is at the centre of the length, we should have

the stem immersed 12 inches and the stern raised 12 inches from the original water line, the sum of these figures equalling the total change.

Moment to Alter Trim One Inch (M').

From the foregoing it will be seen that the total change of trim being known for a given moment, inversely we may get the amount necessary to alter the trim for one inch only, this being a convenient unit with which to calculate changes of trim when a complexity of varying conditions are being dealt with. As we have seen P×1=M1 the moment to change trim, and

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In designing preliminary arrangements of vessels, it is necessary that we should know fairly accurately the moment which it will take to alter the trim one inch (M") to enable us to arrange the principal weights in the ship, and the varying effects on the trim consequent on their alteration in position or removal. For this purpose a close approximation to this moment (M'') is desirable and may be calculated from Normand's formula as follows:

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Where A2=the square of the water plane area, and B=the greatest breadth of water plane. Applying this approximate formula to the foregoing example, we have:

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x .00017259.95 foot-tons,

as against 9.84 foot-tons found by actual calculations, a difference too insignificant to affect noticeably the change in trim.

This moment is useful to have for various draughts, and consequently should be calculated for light and load conditions, and for one or two intermediate spots and a curve of M” run on the usual sheet of "Curves of Elements."

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