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THEORETICAL]
923
SHIPBUILDING


ship acting vertically downwards through its centre of gravity and the resultant pressure of the water on the immersed hull. If the ship be supposed removed and the cavity thus 5g: ', fm formed filled with water, then, since this volume of water is in equilibrium under the same system of fluid pressures, the resultant of these pressures must be equal and opposite to the weight of the water in the cavity and will therefore act vertically upwards through the centre of gravity of this portion of water. Denning the Weight of water displaced by the ship as the displacement, and its centre of gravity as the centre of buoyancy, it is seen that the fundamental conditions for the equilibrium of a ship in still water are (a) that the weight of the ship must be equal to the displacement, and (b) that its centres of gravity and buoyancy must be in the same vertical line. A floating ship is always subject to various external forces disturbing it from its position of equilibrium, and it is necessary to investigate the stability of such a position, i.e. to fizzgfy determine whether the ship, after receiving a small m, ,., ,, m disturbance, will tend to return to its former position, in which case its equilibrium is termed stable, or whether, on the other hand, it will tend to move still farther from the original position, when the equilibrium is termed unstable. The intermediate case, when the ship tends to remain in its new position, is a third state of equilibrium, which is termed neutral. Oi the modes of disturbance possible, it is evident that a bodily movement of the ship in a horizontal direction or a rotation about a vertical axis will not affect the conditions of equilibrium; the equilibrium is also stable for vertical displacements of a ship. The remaining movements, viz. rotations about a horizontal axis, can be resolved into rotations in which the displacement is unaltered, and vertical displacements, the effect of the latter being considered separately. Oi the various horizontal axes about which a ship can rotate two are of particular importance, viz. (1) an axis parallel to the longitudinal plane of symmetry, (2) an axis at right angles to this plane, both axes being so chosen that the displacement remains constant; the stability” of a ship with reference to rotations about these axes is known as the transverse .stability and the longitudinal stability respectively. In the following account the consideration of stability is confined at first to these two cases; the general case of rotation about any horizontal axis whatever being dealt with later.

Trmsvuse b l.et fig. 1 represent ah transverse section of a ship, WL stabmtn emg its water me w en upright, and W L its water line when inclined to a small angle 0 as shown. Assuming that the displacement is unaltered, if G be the position

of the ship's centre of gravity and B, B' the w

A

I

I

Q'

2 G

Y.

W

L

positions of its centre of

buoyanc in the upright

and inclined positions

respectively, the forces

acting on the ship consist

of its weight W

vertically downwards

through G and the resultant

water pressure

equal to W acting vertically

upwards through

B'. These constitute a

U Z coupleof momentW><GZ

where Z is the foot of the

perpendicular from G on

to the vertical througn

B'; the direction of

the couple as drawn in

the figure is such as

would cause the ship

to return to its original

law position, i.e. the equi-FIG

L ibrium is stable for the

inclination shown.

If M be the intersection of the vertical thr ough B' with the original

vertical, the moment of the restoring couple is equal to W XGM sin 9, and GM sin-6 is termed the righting lever. If, by moving weights on board, G be moved to a different position on the original vertical through B, the original position of the ship will remain one of equilibrium, but the moment of stability at the angle of inclination 0 will vary with GM. If G be brought to the position G' above M the moment W><G'Z' will tend to turn the ship away from the original position. It follows that the condition that the original position of equilibrium shall be stable for the given inclination is that the centre of gravity shall be below the intersection of the verticals through the upright and inclined centre of buoyancy; and the moment of stability is proportional to the distance between these two points.

When the inclination 0 is made smaller the point M approaches a definite position, which, in the limit when 0 is indefinitely small, is termed the meta centre.

In ships of ordinary form it is found that for IO to 15 degrees of inclination, the intersection of the verticals through the S IH centres of buoyancy B and B' remains sensibly at the UIZZHOZ; meta centre M; and therefore within these limits the moment of stability is approximately equal to WXGM sin 0. Since the angle on either side of the vertical within which a ship rolls in calm or moderate weather does not usually exceed the limit above stated, the stability and to a great extent the behaviour of a vessel in these circumstances are governed by the distance GM which is known as the rnetacentric height. The position of G can be calculated when the

weights and positions

of the component parts

of the ship are known.

This calculation is

made for a new ship

when the design is

sufficiently advanced

to enable these component

weights and

their positions tovbe determined

with reasonable

accuracy; in the

initial stages of the

design an approximation

to the vertical

position of G is made

by comparison with

previous vessels.

The position of the

centre of gravity of a

ship is entirely independent

of the form or

draught of water, except so far as they affect the amount and distribution of the component weights of the ship. The' position of the meta centre, on the other hand, depends only on the geometrical properties of the immersed part of the ship; and it is determined as follows: ' '

Let WL, W'L' (fig. 2) be the traces of the upright and inclined water planes of a ship on the transverse plane, B, B' the corresponding position of the centre of buoyancy; '0 the angle of inclination supposed indefinitely small in the limit, and S the intersection of WL and W'.L'; join BB'. Q -~~.. Z . " By supposition the displacement is uni changed, and the volumes WAL, W'AL' are equal; on subtracting W'AL it is seen that the two wedges WSW', LSL' are also equalf If dx represent an element of length at right angles to the plane ofytihe figuxie, yi, yz, the half breadths one on each side at any point 'in the original water line, so that WS=y, ,» SL=y2, the areas WSW', LSL' differ, from %y, '.0, %y22 0 by indefinitely small amounts, neglecting which the volumes of WSW', ' LSL' are equal to f%y120dx and f %y226dx. Since these are equal we have

Wu

é yfdx =% yizdx or y1dx><%= y;!lx><;§ ; i.e. the moments of the two portions of the water planeQ about their line of intersection passing through S are equal. This line is also thegaxis of rotation, which therefore passes through the centre of gravity of the water plane. For vessels of the usiial shape, having la middle line plane of symmetry and floating 1n1t1ally upright, for small inclinations consecutive water planes intersect on the middle line.

Again if gl, gg are the centres of gravity of the wedges WSW', LSL', and 11 the volume of either wedge, the moment of transference of the wedges 'v><g1g2 is equal to the moment of transference of the whole immersed volume VXBB' where V is the volume of displacement. But v><g1S=moment of wedge WSW' about 'S = %fy1".0.dx, and vXSg¢=moment of wedge LSL' about S = § fg2?.0.¢lx. Adding, -§ f(y13+y23)0.dx=v><g1g2=V><BB'. But BB'= M.0 to the same order of accuracy, and § f(y13-l-y23) .dx is the moment of inertia of the water plane about the axis of rotation; denoting the latter by I, it follows that BM =I/V; i.e. the height of the meta centre above the centre of buo ancy is equal to the moment of inertia of the water plane about the axis of rotation divided by the volume of displacement. These quantities, 'and also the position of the centre of buoyancy can be obtained by the approximate methods of quadrature usual in ship calculations, and from them the position of the meta centre can be found. g

If the ship is wholly immersed, or if the inertia of the water plane is negligible as in a submarine when diving, BM =O, and the condition for stability is that Q should 'oe below B; the righting lever at any angle of inclination is then equal to BG sin 6. '

During the process of design the position of the centre of gravity