TONS PER INCH
50.4
49.8
GM (LIGHT), 2.99. GM (DEEP), l~ 6T.
DISPLT: IN TONS
15600
11346
13190
8450 7780
the displacement in tons and the number of tons required to increase the mean draught by 1 in., respectively, as ordinates (horizontal). The ordinate of the curve of displacement at any water line is clearly proportional to the area of the curve of tons per inch up to that water line.
The properties of the metacentric stability at small angles are used when determining the vertical position of the centre of gravity Inclining of a ship by an " inclining experiment "; this gives a
expert check on the calculations for this position made in the
meat, initial stages of the design, and enables the stability of the completed ship in any condition to be ascertained with great accuracy.
The experiment is made in the following manner:
Let fig. 6 represent the transverse section of a ship; let w, w be two weights on deck at the positions P, Q, chosen as far apart transversely as convenient ; and let G be the combined centre of gravity of ship and weights. When the weight at P is moved across the deck to Q', the centre of gravity of the whole moves from G to some point G' so that GG' is parallel to PQ' (assumed horizontal) and equal to hw/W where h is the distance moved through by P, and W is the total displacement. The ship in consequence heels to a small angle 0, the new vertical through G passing through the metacentre M ; also GM = GG' cot 8 = hw/W cot 0, the metacentric height being thereby determined and the position of G then found from the metacentric diagram. In practice 0 is observed by means of plumb bobs or a short period pendulum recording angles on a cylinder; 1 the weight w at P, which is chosen so as to give a heel of from 3° to 5°, is divided into several portions moved separately to Q'. The weight at Q' is replaced at P, the angle heeled through again observed; and the weight at Q similarly moved to P' where P'Q=h=PQ', and the angle observed; GM is then taken as the mean of the various evaluations.
In the case of small transverse inclinations it has been assumed that the vertical through the upright and the inclined positions of the Large in centre of buoyancy intersect, or, which is the same thing, c!lnations. that the centre of buoyancy remains in the same trans
verse plane when the vessel is inclined. This assumption is not generally correct for large transverse inclinations, but is nevertheless usually made in practice, being sufficiently accurate
for the purpose of estimating the righting moments and ranges of stability of different ships, calculated under the same conventional system ; this is all that is necessary for practical purposes.
With this assumption, there will always be a point of intersection (M' in fig. 7) of the verticals through the upright and inclined centres of buoyancy; and the righting
lever is, as before,
GZ=GM' sin 0. In this case, however, there is no simple formula for BM' as there is for BM in the limiting case where 0 is infinitesimal; and other methods of calculation are necessary.
The development of this part of the subject was due originally to Atwood, who in the Philosophical Transactions of 1796 and 1798, advanced reasons for differing from the metacentric method which was published by Bouguer in his Traite du navire in 1746. Atwood's treatment of stability (which was the foundation of the modes of calculation adopted in England until about twenty years ago) was as follows:
Let WL, W'L' (fig. 7) be respectively the water lines of a ship when
Such an instrument is described by Froude for recording the " relative " inclination of a ship amongst waves, Transactions of Institution of Naval Architects, 1873, p. 179. The pendulum should have sufficient weight and the arm carrying the pen may be about 4 ft. long. If the cylinder be fitted with a clock recording the time the natural period of the ship will also be obtained.upright and inclined at an angle 0, S their point of intersection: B and B' the centres of buoyancy, gi and g2 the centres of gravity of the equal wedges WSW', L'SL, and hi, h2 the feet of the perpendiculars from gi, g2 on the inclined water line. Draw GZ, BR parallel to W'L', meeting the vertical though B' in Z and R.
The righting lever is GZ as before; if V be the volume of displacement, and v that of either wedge, then
VXBR=vXhih2
GZ=BR—BG sin e; whence the righting moment or
WXGZ=W vXVih2BG sin o .
This is termed Atwood's formula. Since BG, V and W are usually known, its application to the computation of stability at various angles and draughts involves only the determination of v X hih2. A convenient method of obtaining this moment was introduced by F. K. Barnes and published in Trans. Inst. N.A. (1861). The steps in this method were as follows: (a) assume a series of trial water lines at equal angular intervals radiating from S' the intersection of the upright water line with the middle line plane; (b) calculate the volumes of the various immersed and emerged trial wedges by radial integration, using the formula
v = f d0Jr2dx,
0
where r, ¢ are the polar coordinates of the ship's side, measured from S' as origin, and dx an element of length; (c) estimate the moment of transference of the same wedges parallel to the particular trial water line by the formula
vXhih2=1 f ocos(o—o)d4iJrsdx,
0
adding together the moments for both sides of the ship; and (d) add or subtract a parallel layer at the desired inclination to bring the result to the correct displacement. The true water line at any angle is obtained by dividing the difference of volume of the two wedges by the area of the water plane (equal to frdx, for both sides) and setting off the quotient as a distance above or below the assumed water line according as the emerged wedge is greater or less than the immersed wedge. The effect of this " layer correction " on the moment of transference is then allowed.
The righting moment and the value of GZ are thus determined for the displacement under consideration at any required angle of heel.
A different method of obtaining the righting moments of ships at large angles of inclination has prevailed in France, the standard investigation on the subject being that of M. Reech first published in his memoir on the " Construction of Metacentric Evolutes for a Vessel under different Condi
tions of Lading" (1864). The principle of his method is dependent on the following geometrical properties:
Let B', B" (fig. 8) be the centres of buoyancy corresponding to two water lines W'L', WI," inclined at angles 0, o+do, to the original upright water line WL, dO being small; and let
gi, g2 be the centres of
gravity of the equal wedges W'TW", L'TL". The moment of either wedge about the line gigz is zero, and the moments of W'L'A and of W"L"A about gigz
are therefore equal ; since these volumes are also equal, the perpendicular distances of B' and B" from gigs are equal, or B'B" is
parallel to gigs.
The projection on the plane of inclination of the locus of the centre of buoyancy for varying inclinations with constant displacement is termed the curve of buoyancy, a portion BB'B" of which is shown in the figure. On diminishing the angle do indefinitely so that B" approaches B' to coincidence, the line B'B" becomes, in the limit, the tangent to the curve BB'B", and gigs coincides with the water line W'L'; hence the tangent to the curve of buoyancy is parallel to the water line.
Again, if the normals to the curve at B', B" (which are the verticals corresponding to these positions of the centre of buoyancy) intersect at M', and those at B", B" (adjacent to B") at M", and so on, a curve may be passed through M', M' . , commencing at M, the metacentre. This curve, which is the evolute of the curve of buoyancy, is known as the metacentric curve, and its properties were first
also
investigated by Bouguer in his Traite du Navire. The points M'M ... on the curve are now termed prometacentres.
If p represent the length of the normal B'M' or the radius of curvature of the curve of buoyancy at an angle 0, then p.dO = ds the length of an element of arc of the B curve. In the limit when dB is
indefinitely small, =p. Using Cartesian coordinates with B as
origin and By, Bz, as horizontal and vertical axes, we have
d8 =5 cos B=p cos 0, (I)
dzds (2)
dB _dB sin B = p sin B;
whence
e e
y= f p. cos B.dO; z= p. sin B.dO,
and the righting lever GZ=y cos B+(z—BG) sin B.
The radius p is (as for the upright position) equal to the moment of inertia of the corresponding waterplane about a longitudinal axis through its centre of gravity divided by the volume of displacement; the integration may be directly performed in the case of bodies of simple geometrical form, while a convenient method of approximation such as Simpson's Rules is employed with vessels of the usual shipshaped type. As an example in the case of a box, or a ship with upright sides in the neighbourhood of the waterline, if BG=a and BM =po, then p=po sec3 0;
whence
e
y = o p cos 0. dO = po tan 0,
e
=f p sin B.dO ='lpo tang 0,
GZ=(po—a) sin B+zpo tang B. sin 0;
which relations will also hold for a prismatic vessel of parabolic section. It is interesting to note that in these cases if the stability for infinitely small inclinations is neutral, i.e. if po=a, the vessel is stable for small finite inclinations, the righting lever varying approximately as the cube of the angle of heel.
The application of the preceding formulae to actual ships is troublesome and laborious on account of the necessity for finding by trial the positions of the inclined waterlines which cut off a constant volume of displacement. To avoid this difficulty the process was modified by Reech and Risbec in the following manner:—Multiply equations (1) and (2) by V.dO, V being the volume of displacement; we then have
d(Vy)=I cos B. dB, (3)
d(Vz) =I sin 0.dO, (4)
where I is the moment of inertia of the inclined waterline about a longitudinal axis passing through its centre of gravity. These formulae have been obtained on the supposition that the volume V is constant while 0 is varying; but by regarding the above equations as representing the moments of transference horizontally and vertically due to the wedges, it is evident that V may be allowed to vary in any manner provided that the moment of inertia I is taken about the longitudinal axis passing through the intersection of consecutive waterlines. In particular the waterlines may 'all be drawn through the point of intersection of the upright waterline with the middle line, and the moments of inertia are then equal to ifr3dx for both sides of the ship, r being the halfbreadth along the inclined waterline; the increase in volume is the difference between the quantity fd8fir2dx for the two sides of the ship.
If V°, Vo be the volumes of displacement at angles a and o respectively,
Va—Vo=fay[ f differ ce]' (5)
and substituting in (3) and (4) and integrating,
vay = fo dB [f 3sumx] cos 0, . (6)
0
Vaz =f adB [ J 3sumx] sin B. . (7)
0
On eliminating Vain (5), (6) and (7), y and z can be found.
This is repeated at different draughts, and thus V., y and z are determined at a number of draughts at the same angle, enabling curves of y and z to be drawn at various constant angles with V for an abscissa; from these, curves may be obtained for y and z with the angle a as abscissa for various constant displacements; GZ being equal to
y cos a+(z—a) sin a.
From the foregoing it is evident that the elements of transverse stability, including the coordinates of the centre of buoyancy, positionof prometacentre, values of righting lever and righting moment, depend on two variable quantities—the displacement and the angle of heel. The righting lever GZ is in England selected curve of as the most useful criterion of the stability, and, after curve it
being evaluated for the various conditions, is plotted
in a form of curves—(a) for various constant displacements on an abscissa of angle of inclination, (b) for a number of constant
w
W z
N
L7
angles on an abscissa of displacement. These are known as curves of stability and cross curves of stability respectively; either of these can be readily constructed when the other has been obtained; which process is utilized in the method now almost universally adopted for obtaining GZ at large angles of inclination, a full description being given in papers by Merrifield and AmsIer in Trans. I.N.A. (188o and 1884). The procedure is as follows:
1. The substitution of calculations at constant angle for those at constant volume. A number of waterlines at inclinations having a constant angular interval (generally 15°) are drawn passing through the intersection S' of the load waterline with the middle line on the body plan. Other waterlines are set off parallel to these at fixed distances above or below the original waterline passing through S'.
2. The volumes of displacement and the moments about an axis through S' perpendicular to the waterline are determined for each draught and inclination by means of the AmslerLaffon integrator,
the pointer of this instrument being taken in turn round the immersed part of each section.
3. On dividing the moments by the corresponding volumes, the perpendicular distance of the centre of buoyancy from the vertical through S' is obtained, i.e. the value of GZ, assuming G and S' to coincide.
4. For each angle in turn " cross curves " of GZ are drawn on a base of displacement.
and
5. From the cross curves, curves of stability on a base of angle of inclination can be constructed for any required displacement, allowance being made for the position of G by adding to, or subtracting from, each ordinate, the quantity GS' sin a according as G is below or above S'.
A typical set of cross curves of stability for a battleship of about 18,000 tons displacement is shown in fig. 9. It will be observed that the righting levers decrease with an increase of displacement ; and this is a general characteristic of the cross curves for ships of ordinary
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