her move away from the upright position. This matter will be more fully explained hereafter. Starting from the upright, a ship may be inclined transversely, or longitudinally, or in any "skew" direction lying between the two. It is only necessary, however, to consider transverse and longitudinal inclinations in connection with statical stability; the innumerable possible skew inclinations being easily dealt with when the conditions of stability for the two principal inclinations have been ascertained. The minimum stability of a ship corresponds to transverse inclinations; the maximum stability, to longitudinal inclinations. It is, therefore, of the greatest importance to thoroughly investigate the changes in the statical stability of ships as they are heeled to greater and greater transverse inclinations, especially for ships which have masts and sails. Longitudinal stability is less important, but claims some notice, especially as regards its influence on changes of trim and pitching Taking first transverse inclinations, let them be supposed to be small; it is then easy to estimate the statical stability when the position of the metacentre is known. For our present purpose the metacentre may be defined, with sufficient exactitude, as the intersection (M in the cross-section, Fig. 30) of the line of action (BM) of the buoyancy when the ship is inclined through a very small angle, with the line of action (B,GM) of the buoyancy when the ship is upright and at rest. In vessels of ordinary form, no great error is introduced by supposing that, for angles of inclination between the upright and 10 or 15 degrees, all the lines of action of the buoyancy (such as BM) pass through the same point (M)— the metacentre. For any angle of inclination a within these limits the perpendicular distance (GZ) of the line of action of the buoyancy from the centre of gravity is determined by— GZ = GM sin a. Moment of statical stability = D × GM sin a. As an example, take a ship weighing 6000 tons, for which the distance GM = 3 feet, and suppose her to be steadily heeled under canvas at an angle of 9 degrees. Then Moment of statical stability = 6000 tons × 3 feet × sin 9° = 18,000 × 1564 2815 foot-tons. = For most ships the angles of steady heel under canvas lie within the limits for which the metacentric method holds; and consequently this method may be used in estimating the "stiffness" of a ship, i.e. her power to resist inclination from the upright by the steady pressure of the wind on her sails. It must be noticed that this term "stiffness" is used by the naval architect in a sense distinct from "steadiness." A stiff ship is one which opposes great resistance to inclination from the upright, when under sail or acted upon by some external forces; a crank ship is one very easily inclined; the sea being supposed to be smooth and still. A steady ship, on the contrary, is one which, when exposed to the action of waves in a seaway, keeps nearly upright, her decks not departing far from the horizontal. Hereafter it will be shown that frequently the stiffest ships are the least steady, while crank ships are the steadiest in a seaway. At present we are dealing only with still water, and must limit our remarks to stiffness. From the foregoing remarks it will be evident that, so far as statical stability is concerned, and within the limits to which the metacentric method applies, a ship may be compared to a pendulum, having its point of suspension at the F metacentre (M, Fig. 30), and its weight concentrated in a "bob" at the centre of gravity G. Fig. 31 shows such a pendulum, inclined to an angle a. The weight (D) acting downwards produces a tendency to return to the upright, measured by the moment Dx GM sin a, which is identical with the expression for the righting moment of the ship at the same angle. But this comparison holds only while the ship and the pendulum are at rest; as soon as motion begins, the comparison ceases to be correct, and the failure to distinguish between the two cases has led some writers into serious error. Changes in the height (GM) of the metacentre above the centre of gravity produce corresponding changes in the stiffness of a ship; in fact, the stiffness may be considered to vary with this height-usually termed the " metacentric height." If it is doubled, the stiffness is doubled; if halved, the stiffness is reduced by one-half, and so on. Care has, therefore, to be taken by the naval architect, in designing ships, to secure a metacentric height which shall give sufficient stiffness, without sacrificing steadiness in a sea-way. In adjusting these conflicting claims, experience is the best guide. The following tables contain particulars of the metacentric heights of different classes of war-vessels belonging to the Royal Navy or to foreign navies; the vessels being fully laden. Ironclads. 1. Converted frigates (formerly two-deckers); Prince) Consort class in Royal Navy, and earliest French frigates (Gloire class) 2. Warrior and Minotaur classes in Royal Navy; Flandre class in French navy 3. Recent types of frigates, such as Bellerophon, Hercules, or Alexandra in Royal Navy 4. Marengo class (last completed) in French navy 5. Alma class of corvettes in French navy 6. Devastation class of Royal Navy 7. Glatton (low freeboard monitor) 8. Garde-côtes (Bélier class), French navy 9. American type of monitor (Miantonomoh) It is to be noted that the first five groups in this table include sailing ironclads. Experience has led to the selection of metacentric heights of from 3 to 4 feet as the best suited for such vessels, taking into account their ordinary spread of canvas. The remaining groups comprehend mastless ships, in which the greater metacentric heights are often unavoidable with the forms and proportions rendered necessary by the special conditions of the designs-such as moderate draught in association with thick armour and heavy guns. Unarmoured Ships. 1. Screw line-of-battle ships (two-deckers), of which a few remain in the French and Royal navies 2. Screw frigates and corvettes of the old types 4. Screw corvettes and sloops of recent design. 5. Smaller classes of sea-going vessels 6. Tugs and small vessels, not sea-going 31 24 11 2 When the consumable stores of these vessels, armoured and unarmoured, are removed, the metacentric heights are commonly about one foot less than in the fully laden condition to which the tables refer. For merchant ships, corresponding particulars are not on record, few experiments having been made to determine them; moreover, in these vessels variations in stowage of cargo must produce considerable variations in the metacentric heights. There is reason to believe that recent merchant steamers, having extreme proportions of length to breadth, have metacentric heights much less than those stated above for war-ships. The naval architect usually has far greater control over the vertical position of the metacentre in a newly designed ship than he has over that of the centre of gravity. In a warship the distribution of the armour, armament, and equipment are settled mainly with reference to fighting efficiency; and this distribution chiefly controls the vertical position of the centre of gravity. Merchant ships have to fulfil specified conditions as to draught, freeboard, and carrying power, besides being subject to variations in the character and stowage of the cargoes; and these variations may produce considerable changes in the vertical position of the centre of gravity, even when the total loads carried are identical on different voyages. On the other hand, the position of the metacentre in a ship depends only on her form, and the extent to which she is immersed; it is quite independent of the structural arrangements or the lading. Two ships of identical form, immersed to the same extent, and therefore having equal displacements, may have the equal weights carried so differently disposed that the centre of gravity in one will be considerably higher than that in the other; but the metacentres will occupy the same position in both vessels. By means of changes in the form of the water-line section and displacement, in the proportions of length to breadth, or in draught of water, the designer can, however, associate with a constant total weight, or displacement, very various vertical positions of the metacentre. It has been explained that the metacentre affords a ready means of determining the line of action of the buoyancy for any inclined position, and avoiding the necessity for determining the place of the corresponding centre of buoyancy. But in practice the position of the metacentre is fixed with reference to the centre of buoyancy, corresponding to the upright position of the ship. The distance (B,M, Fig. 30) is given by the formula,* B1M = Moment of inertia of water-line area The "moment of inertia" of an area may be defined as the sum of products of each element of that area, by the square of its distance from the axis about which the moment of inertia is to be calculated. The proof of the formula given above involves mathematical treatment which would be out of place here. |