THEORETICAL SHIPBUILDING Stability.—A ship floating at rest in still water is in the same condition as any other floating body and the rules of elementary hydrostatics must be satisfied. These are that (I) the weight of the ship must equal the weight of water displaced known as the displacement: and (2) the centre of gravity of the ship and the centre of gravity of the water displaced—known as the centre of buoyancy—must be in the same vertical line. It is further necessary that the ship upon being slightly disturbed from its position of equilibrium, should tend to return to that position, when the equilibrium is described as stable. Should the ship on receiving such a disturbance tend to move still farther the equilibrium is unstable : and the intermediate case where she tends to remain in the new position is described as a condition of neutral equilibrium.
Of the various modes of disturbance to which a ship may be subjected it is sufficient for practical purposes only to consider that caused by rotation about a horizontal axis, the displacement remaining unaltered. Such a rotation can in general be resolved by the ordinary statical rules into two rotations about perpendicu lar axes and it is convenient to consider the question with refer ence to (I) an axis parallel to the fore and aft M.L. plane of the ship and (2) an axis at right angles to this plane; these rotations result in stability conditions known as transverse stability and longitudinal stability respectively.
As drawn, the couple tends to restore the ship to the upright and consequently the equilibrium is stable. If G were above M the couple would tend to capsize the ship and the equilibrium would be unstable.
Hence for stability of equilibrium the centre of gravity must be below the point M, and the moment of stability is proportional to GM. When 0 is indefinitely small the point M tends to a fixed limit and is then termed the metacentre, the distance GM being known as the metacentric height. It is found that in ships of ordinary form for inclinations up to approximately 15° the point of intersection of the vertical is through the points B and B' coincides very nearly with the metacentre, so that within these limits the moment of stability is approximately W X GM sin 0.
The position of the centre of gravity can be calculated when the weights and positions of the several parts are known. The position of the metacentre de pends entirely on the geometry of the immersed portion of the ship. A simple analytical investi gation shows that the axis of rota tion must pass through the centre of gravity of the water plane— which point is known as the cen tre of flotation—and that the height of the metacentre above the centre of buoyancy is equal to the moment of inertia of the water plane about the axis of rotation divided by the volume of displacement; with the usual notation, i.e., BM= • These quantities, together with the position of the centre of buoyancy are found by the ordinary methods adopted for ship calculations and the position of M in the ship found. Thus by selecting the underwater form any desired value of GM can be obtained ; such value differs in various types of ship, experience determining the desirable figure for each type.
Typical values of metacentric height for various classes of ships are as follows :— Changes in metacentric height are caused by alteration in dis placement or in the position of the centre of gravity, and it is necessary to investigate the stability in several conditions. The position of the centre of buoyancy and metacentre are readily calculated for several W.L's which correspond to calculable dis placements ; and the position of G for any given condition can be calculated. The results are conveniently shown on a metacentric diagram in which the curves of height of metacentres and vertical positions of the centres of buoyancy are set up from a line inter secting the W.L's at Inclining calculation of the position of G is a lengthy and laborious process and is inevitably of a more or less approximate nature. When the ship is nearly complete a check is obtained on the position of G by use of the properties of metacentric stability. This is termed an inclining experiment and is carried out by moving a known weight (w) through a known transverse distance (d), thereby producing a couple tending to in cline the ship. This couple is resisted by the stability couple of the ship and a position of equilibrium is reached at some angle 0 which is directly measured by pendulums. We then have The GM in the inclining condition is thus known at once : the position of M is known from the metacentric diagram and con sequently the vertical position of G for this condition determined.