Theoretical Shipbuilding

The position of G in any other condition is readily obtained by a simple process of algebraic additions.

Large Inclinations.

The intersection of the two verticals through the centres of buoyancy in the upright and the inclined condition is the fixed point M only when the inclination is small.

For larger inclinations it becomes necessary to investigate the mat ter further. For all practical purposes it is sufficient to assume that the centre of buoyancy re mains in the same transverse plane, when the ship is inclined. The conditions are as indicated in fig. 2 and the righting lever is GZ = GM' sine. The point M' is known as the pro-metacentre and it is important to note that in general its position alters with the angle of inclination. The problem now is to determine the value of BM', for which there is no simple formula.

If v be the volume of either wedge and gi, g2 their respective centres of gravity: Then V X BR=vXhih2.

Also GZ=BR—BG sine whence the righting moment of the curve of flotation, which is the envelop of all possible water lines for the ship when inclined transversely at constant displace ment.

Dynamical Stability.

The dynamical stability of a ship at any angle is defined as the work done in inclining the ship from the upright. Its value is conveniently obtained by integrating the curve of stability, i.e., dynamical stability = W X f GZ.c1 0, and a curve of dynamical stability on a base of angles is readily obtained.

Sailing Ships.

A ship with sails is subjected to the wind pressures on the sails and upper works and to the water pressures on the hull. The wind pressure on the sails is dependent on the wind velocity and on the form and area of the sails ; and the amount of sail that can safely be carried is contingent upon the existence of large statical stability at moderate angles of heel.

The stability of a sailing vessel is usually estimated by assum ing all plain sail to be placed in a f ore and aft direction and to be subject to a normal pressure of i lb. per sq.ft. corresponding to a wind of about 16 knots. The resultant pressure of the wind is taken as acting through the C.G. of the total sail area, this point being known as the centre of effort. The resultant pressure of the water on the hull is assumed to pass through the centre of gravity of the immersed M.L. plane : this point is known as the centre of lateral resistance. Then if h be the distance between these points in feet, A the sail area in sq. ft. and a the angle of heel, then the following equation is approximately true.

This is known as Atwood's formula. Thus the value of the right ing moment at any angle resolves itself into evaluating the ex pression v X hih2. Several methods can be adopted, one of the most convenient being that introduced by Barnes and published in Trans. Institute of Naval Architects 1861.

A different method of approaching the problem has generally been adopted in France; the investigation being due to Reech and first published in his memoir "Construction of Metacentric Evolutes for a Vessel under different Conditions of Lading" (1864). These methods enable GZ to be obtained for a fixed displacement for any angle and the curve obtained by plotting these values as ordinates on a base of angles furnishes a curve of stability and a similar curve can be obtained for any desired dis placement.

All such methods are cumbersome, and the method now uni versally adopted for obtaining GZ at large angles of inclination is that described in papers by Merrifield and Amsler in Trans. Insti tute of Naval Architects (188o and 1884). With this method in which the work can be expeditiously carried out by the use of the integrator, the value of GZ is obtained for a number of con stant angles at varying displacements. The curve obtained for a fixed angle by plotting GZ as ordinates on a base of displacement is known as a cross curve of stability : and a similar curve can be obtained for any desired angle. It is obvious that either set of curves can be readily constructed when the other is available. In carrying out these calculations it is necessary to make certain assumptions, the principal of which are that all openings in weather deck and in ship's side up to the weather deck are covered in and made watertight and that all weights in the ship are fixed. The slope of the curve of stability for small angles, the maxi mum value of GZ and the angle at which it occurs, and the angle at which GZ vanishes and upon reaching which the ship is about to capsize, determine the character of the curve. Other important factors influence the curve, viz., the freeboard, an increase in which tends to lengthen-out the curve and thus make the vanish ing angle larger; and the position of C.G. which affects both the initial stability and the range. To sum up, initial stability is de pendent essentially on beam and position of C.G. while range is dependent essentially on freeboard and position of C.G.