Page:EB1911 - Volume 24.djvu/991

centre of gravity, or, as it is termed, the centre of flotation, the curve of flotation will be the locus of the projections of the centres of flotation on the plane of the figure, which curve touches each waterline.

From consideration of the slope of a ship's side around the periphery of a water-line, Dupin obtained the following expression for ρ, the radius of curvature of the curve of flotation,

ρ′ ＝ fy² tan α. ds/area of water-planefor both sides,

where ds is an element of the perimeter, α the inclination of the ship's side to the vertical, and y its distance from the longitudinal axis through the centre of flotation. M. Emile Leclert, in a paper read at the Institution of Naval Architects, 1870, proved the e uivalence of the above formula to the two following, which are (known as Leclert's Theorem:

ρ′＝ρ+Vdρ/dV and ρ′＝dI/dV,

where I and V are respectively the moment of inertia of the water plane and the volume of displacement, and ρ is the radius of the curve of buoyancy or B′M′. Independent analytical proofs of the formulae were given in the paper referred to; and (Trans. I.N.A., 1894) a number of elegant geometrical theorems in connexion with stability, given by Sir A. G. Greenhill, include a demonstration of Leclert's Theorem as follows (in abbreviated form):

Let B, B₁ (fig. 17) be the centres of buoyancy of a ship in two consecutive inclined positions, and F, F, the corresponding centres of flotation. Draw normals BM, BIM, meeting at the pro-meta centre M, and FC, FiC, meeting at the centre of curvature C. Produce FB, F1B, to meet at O; join OM, MC.

Then BM, CF and B1M, CF, are respectively parallel, and ultimately also BB₁ , FF₁, ; hence the triangles MBB₁, CFF₁ are similar and

BM/CF＝BB₁/FF₁＝OB/OF,

so that O, M and C are collinear.

Fig. 17.

If the displacement V be now increased by dV, changing B to B′, and M to M′, then since the added displacement dV may be supposed concentrated at F, B' will lie on OBF, and it may be shown similarly as before that M' lies on OC. Further, considering the transference of moments, BB′×V=BF×dV.

Draw MED parallel to BF, then

dV/V＝BB′/BF＝ME/MD＝M′E/CD＝dρ/ρ′−ρ;

∴dρ/dV＝ρ′−ρ/V or ρ′＝+Vdρ/dV,

giving Leclert's first expression; also, since ρ = I/V,

dI/dV＝ρ+Vdρ/dV＝ρ′,

which is Leclert's second expression for p'.

The value of ρ′ at the upright can be obtained from the meta centric diagram by the following simple construction. Let M and B be the meta centre and the centre of buoyancy for a water-line WL on the metacentric diagram (fig. 18); draw the tangent to the B curve meeting WL at Q, and through Qdraw QR to meet MB and parallel to the tangent to the M curve at M.

Let BP=h, and area of water-line be A. Then

PQ＝h cot θ＝hV/Ah＝V/A;

also

MR＝BM−(BP+PR)＝ρ−V/A (tan θ+tan φ).

If D be the draught,

tan θ+tan φ ＝dρ/dD ＝ −A.dρ/dV,

whence

MR＝ρ+Vdρ/dV＝ρ′

the curve of flotation being concave upwards if R is below M.

For moderate inclinations from the upright, the buoyancy of the added layer due to a small additional submersion will act through the centre of curvature of the curve a of flotation; this point may be regarded as that at which any additional weight will, onf being placed on a ship, cause no difference to the values of the righting moment at moderate angles of inclination. The curve of flotation, therefore, and its evolute bear similar relations to the increase or decrease of the stability of a ship due to alteration of draught, as the curves of buoyancy and of pro-metacentres do to the actual amount of the stability.

The curve of flotation resembles the curve of buoyancy in that not more than two tangents can be drawn to it in any given direction, but it differs in that its radius of curvature can become infinite or change sign. It contains a number of cusps determined by ρ′＝dI/dV＝O. These occur in an ordinary ship-shape body at positions: (1) at or near the angles at which the deck is immersed or emerged (four in number); and (2) at or near the angles 90° and 270°. There are, therefore, six cusps in the curve ¢ of flotation of an ordinary ship; they are shown in figs. 15 and 16 by the points F₂, F₃, F₄, F₆, F₇, F₈.

The following relations between the curves of buoyancy and of pro-meta centres and the curve of statical stability are of interest, and enable the former curves to be constructed when the latter have been obtained. If GZ′, GZ″ (fig. 19) are the righting levers correspond in to inclinations θ, θ+ dθ, where dθ vanishes in the limit; B′, B″, the centres of buoyancy, metacentre; produce GZ' to meet, B″M′ in U.

Then, neglecting squares of small quantities,

d(GZ′)＝Z′U＝M′Z′.dθ,

or vertical distance of M′ above G＝d(GZ′)/dθ.

Also M′B′=M′B″;

hence

Z″B″−Z′B′＝MZ′-MZ″＝Z″U＝GZ′.dθ,

or

GZ＝B′Z′/dθ,

i.e. the vertical distance (B′Z′) of G over B is equal to ∫GZ.dθ.

It follows that by differentiating the levers of statical stability and finding the slope at each ordinate the vertical distance of M′ over G is obtained, and M′ may be plotted by setting up this value from Z′ above GZ′ drawn at the correct inclination; also that by integrating the curve of statical stability and finding its area up to any angle, the vertical separation of G and B′ is obtained, and B′ may be plotted by setting down this value increased by BG below Z′.