Page:EB1911 - Volume 24.djvu/987

the displacement in tons and the number of tons required to increase the mean draught by 1 in., respectively, as ordinates (horizontal). The ordinate on the curve of displacement at any water line is clearly proportional to the area of the curve of tons per inch up to that water line.

The properties of the metacentric stability at small angles are used when determining the vertical position of the centre of gravity of a ship by an “inclining experiment”; this gives a check on the calculations for this position made in the initial stages of the design, and enables the stability of the completed ship in any condition to be ascertainedInclining experiment. with great accuracy.

The experiment is made in the following manner:—

Let fig. 6 represent the transverse section of a ship; let 𝑤, 𝑤 be two weights on deck at the positions P, Q, chosen as far apart transversely as convenient; and let G be the combined centre of gravity of ship and weights. When the weight at P M is moved across the deck to Q′, the centre of gravity of the whole moves from G to some point G′ so that GG' is parallel to PQ′ (assumed horizontal) and equal to ℎ𝑤/W where ℎ is the distance moved through by P, and W is the total displacement. The ship in consequence heels to a small angle θ, the new vertical through G passing through the metacentre M; also GM＝GG′ cot θ＝ℎ−𝑤/W cot θ, the metacentric height being thereby determined and the position of G then found from the metacentric diagram. In practice θ is observed by means of plumb bobs or a short period pendulum recording angles on a cylinder;^[1] the weight 𝑤 at P, which is chosen so as to give a heel of from 3° to 5°, is divided into several portions moved separately to Q′. The weight at Q′ is replaced at P, the angle heeled through again observed; and the weight at Q similarly moved to P' where P′Q＝ℎ＝PQ′, and the angle observed; GM is then taken as the mean of the various evaluations.

Fig. 6.

In the case of small transverse inclinations it has been assumed that the vertical through the upright and the inclined positions of the centre of buoyancy intersect, or, which is the same thing, that the centre of buoyancy remains in the same transverse plane when the vessel is inclined. This assumptionLarge inclinations. is not generally correct for large transverse inclinations, but is nevertheless usually made in practice, being sufficiently accurate for the purpose of estimating the righting moments and ranges of stability of different ships, calculated under the same conventional system; this is all that is necessary for practical purposes.

With this assumption, there will always be a point of intersection (M′ in fig. 7) of the verticals through the upright and inclined centres of buoyancy; and the righting lever is, as before, GZ＝GM′ sin θ. In this case, however, there is no simple formula for BM′ as there is for BM in the limiting case where θ is infinitesimal; and other methods of calculation are necessary.

The development of this part of the subject was due originally to Atwood, who in the Philosophical Transactions of 1796 and 1798, advanced reasons for differing from the metacentric method which was published by Bouguer in his Traité du navire in 1746. Atwood's treatment of stability (which was the foundation of the modes of calculation adopted in England until about twenty years ago) was as follows:—

Let WL, W′L′ (fig. 7) be respectively the water lines of a ship when

Fig. 7.

upright and inclined at an angle θ, S their point of intersection: B and B′ the centres of buoyancy, 𝑔₁ and 𝑔₂ the centres of gravity of the equal wedges WSW′, L′SL, and ℎ₁, ℎ₂ the feet of the perpendiculars from 𝑔₁ and 𝑔₂ on the inclined water line. Draw GZ, BR parallel to W'L', meeting the vertical through B′ in Z and R.

The righting lever is GZ as before; if V be the volume of displacement, and 𝑣 that of either wedge, then

V × BR＝𝑣 × ℎ₁ℎ₂

also

GZ＝BR−BG sin θ;

whence the righting moment or

W×GZ＝W$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$𝑣×ℎ₁ℎ₂/V −BG sin θ$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ .

This is termed Atwood's formula. Since BG, V and W are usually known, its application to the computation of stability at various angles and draughts involves only the determination of 𝑣×ℎ₁ℎ₂. A convenient method of obtaining this moment was introduced by F. K. Barnes and published in Trans. Inst. N.A. (1861). The steps in this method were as follows: (a) assume a series of trial water lines at equal angular intervals radiating from S′ the intersection of the upright water line with the middle line plane; (b) calculate the volumes of the various immersed and emerged trial wedges by radial integration, using the formula

𝑣＝1/2$\int _{0}^{\theta }$𝑑φ∫𝑟²dx,

where 𝑟, φ are the polar co-ordinates of the ship’s side, measured from S′ as origin, and dx an element of length; (𝑐) estimate the moment of transference of the same wedges parallel to the particular trial water line by the formula

𝑣×ℎ₁ℎ₂＝1/3$\int _{0}^{\theta }$cos(φ−φ)𝑑φ∫𝑟³𝑑𝑥,

adding together the moments for both sides of the ship; and (𝑑) add or subtract a parallel layer at the desired inclination to bring the result to the correct displacement. The true water line at any angle is obtained by dividing the difference of volume of the two wedges by the area of the water plane (equal to ∫𝑟𝑑𝑥, for both sides) and setting off the quotient as a distance above or below the assumed water line according as the emerged wedge is greater or less than the immersed wedge. The effect of this “layer correction” on the moment of transference is then allowed.

The righting moment and the value of GZ are thus determined for the displacement under consideration at any required angle of heel.

A different method of obtaining the righting moments of ships at large angles of inclination has prevailed in France, the standard investigation on the subject being that of M. Reech first published in his memoir on the “Construction of Metacentric Evolutes for a Vessel under different Conditions of Lading” (1864). The principle of his method is dependent on the following geometrical properties:—

Let B′, B″ (fig. 8) be the centres of buoyancy corresponding to two water lines W′L′, W″L″ inclined at angles θ, θ+𝑑θ, to the original upright water line WL, 𝑑θ being small; and let 𝑔₁, 𝑔₂ be the centres of gravity of the equal wedges W′TW″, L′TL″. The moment of either wedge about the line 𝑔₁𝑔₂ is zero, and the moments of W′L′A and FIG. 8. of W″L″A about 𝑔₁𝑔₂ are therefore equal; since these volumes are also equal, the perpendicular distances of B′ and B″ from 𝑔₁𝑔₂ are equal, or B′B″ is parallel to 𝑔₁𝑔₂.

The projection on the plane of inclination of the locus of the centre of buoyancy for varying inclinations with constant displacement is termed the curve of buoyancy, a portion BB′B″ of which is shown in the figure. On diminishing the angle 𝑑θ indefinitely so that B″ approaches B′ to coincidence, the line B′B″ becomes, in the limit, the tangent to the curve BB′B″, and 𝑔₁𝑔₂ coincides with the water line W′L′; hence the tangent to the curve of buoyancy is parallel to the water line.

Again, if the normals to the curve at B′, B″ (which are the verticals corresponding to these positions of the centre of buoyancy) intersect at M′, and those at B″, B″′ (adjacent to B″) at M″, and so on, a curve may be passed through M′, M″, . . . , commencing at M, the metacentre. This curve, which is the evolute of the curve of buoyancy, is known as the metacentric curve, and its properties were first

1. Such an instrument is described by Froude for recording the “relative” inclination of a ship amongst waves, Transactions of Institution of Naval Architects, 1873, p. 179. The pendulum should have sufficient weight and the arm carrying the pen may be about 4 ft. long. If the cylinder be fitted with a clock recording the time the natural period of the ship will also be obtained.