An excellent approximate formula for the longitudinal B.M. is given by J. A. Normand in the 1882 transactions of the I.N.A. Taking the symbols we have been using : 42 X I L.B.M. = .0735 Вхү Applying this formula to the vessel with which we are dealing, we find : 832.142 x 100 L.B.M. =.0735 = 164.12 feet. 12 X 2583.7 which is a very close approximation to the calculated result of 165 feet. We may also use the approximate formula which we applied in the case of the transverse B.M. altered to suit the new axis with a modified coefficient, as : L.B.M. = L3 X B X ii. Moment to Change Trim (Mi). As the centre of gravity of the displacement (or centre of buoyoncy), either in the vertical or the longitudinal direction may be an entirely different locus from the ship's centre of gravity, it is obvious that unless the moment of the weights of the ship and engines, with all equipment weights, balances about the centre of buoyancy we shall have a preponderating moment deflecting the head or stern, as the moment is forward or aft of the C.B., respectively, until the vessel shall have reached a trim in which the pivotal point or C.B. is in the same vertical line as the completed ship's centre of gravity. To determine the moment necessary to produce a change of trim (Mi) in a given ship, it is necessary to know the vertical position of the centre of gravity of the vessel and the height of the longitudinal metacentre (L.M.C.). The former may be calculated in detail or preferably proportioned from a similar type ship whose centre of gravity has been found by experiment; i although great accuracy in the location of this centre in calculat ing the moment is not as important as in the case of G. M. for initial stability, as small variations in its position can only affect the final result infinitesimally. To investigate the moment affecting the trim, let us move a weight P already on board of the 100foot steamer whose calculations are being figured. D= Weight of ship including weight P = 73.82 tons. P= 5 Tons. 1 = 50 feet (distance moved). L= 100 feet (length of vessel). e In the figure we have the centre of gravity G to Gı, and the centre of buoyancy from B to B1, due to the shifting of the weight P forward for a distance represented by l, giving a moment PXT D The new water line is at WiLi and B1G1 are in the same vertical and at right angles to it, and the point of intersection of the original and new water line at “O” equal to the centre of gravity (flotation) of water plane, therefore the triangles GMG1, WOW, and LOL1, are of equal angle, so that GG1 WW1_LL LLL WW1 + LLI. L, then WW, + LLI; WO+LO but we have seen that GM X change of trim PXL D PXL XL feet. D X GM 2. 116 feet = 24] inches. D X GM 73.82 x 160 Calling this change of trim 24 inches, and assuming that the point of intersection “0” is at the centre of the length, we should have the stem immersed 12 inches and the stern raised 12 inches from the original water line, the sum of these figures equalling the total change. Moment to Alter Trim One Inch (M"). From the foregoing it will be seen that the total change of trim being known for a given moment, inversely we may get the amount necessary to alter the trim for one inch only, this being a convenient unit with which to calculate changes of trim when a complexity of varying conditions are being dealt with. As we have seen Pxl=M; the moment to change trim, and M1 XL D X GM therefore, D X GM 1 foot or one inch M". 12 x L Substituting values we have : feet ; 73.82 X 160 M' = = 9.84 foot-tons. 12 X 100 In designing preliminary arrangements of vessels, it is necessary that we should know fairly accurately the moment which it will take to alter the trim one inch (M'') to enable us to arrange the principal weights in the ship, and the varying effects on the trim consequent on their alteration in position or removal. For this purpose a close approximation to this moment (M'') is desirable and may be calculated from Normand's formula as follows: A2 o'ra x 30.9 B Where A2=the square of the water plane area, and B=the greatest breadth of water plane. Applying this approximate formula to the foregoing example, we have: 832.142 x .0001725 = 9.95 foot-tons, 12 as against 9.84 foot-tons found by actual calculations, a difference too insignificant to affect noticeably the change in trim. This inoment is useful to have for various draughts, and consequently should be calculated for light and load conditions, and for one or two intermediate spots and a curve of M" run on the usual sheet of " Curves of Elements." M"= Alteration in Trim through Shipping a Small Weight. If it be required to place a weight on board but to retain the same trim, i.e., to float at a draught parallel to the original one, the weight added must be placed vertically above the centre of gravity of the water plane. Should, however, the weight be required in a definite position, then the altered trim will be as under : Instead of dealing with the weight at P let us assume firstly that it is placed on board immediately over the C.G. of water plane, when we shall find the parallel immersion to be a layer P equal to the distance between WL and WiL1 whose depth is " Let the weight be now moved to its definite position at a distance I forward of C.G., then PXL,XL =C. (D + P) GM GM of course will be the amended height due to altered condition after the addition of P. Then : Of course we assume that the alteration is of like amount forward as aft. This is only partly correct, but where small weights are dealt with is sufficiently so for most purposes. Generally the ship is fuller aft on and near the load line than forward, and probably a water plane midway between base and L.W.L. would have its centre of flotation at the half length, so that a curve drawn through the centres of gravity of the water planes would incline aft, and as we have assumed the weight as being placed on board over the C.G. the original water plane, is ob us at the new line will have its centre of flotation somewhat further aft, and consequently the tangent of the angle WOW, will be less than that of LOL2. With large weights and differences in the two draughts, the disparity would become sufficiently great to require reckoning, in which event the assumed parallel line in the preceding case would give the water line from which to determine the centre of flotation. Thereafter on finding the change of trim, which we shall call 10 inches, the amount of immersion of stem and emersion of stern post would be in proportion to the distance from 0 to stem and O to post relatively to the length of water line. If we call “0” to stem 60 feet and “0” to post 40 feet, the water line length being 100 feet, we have : Immersion forward 10"=6 inches | Total change 10 inches. TCHIBYSCHEFF'S RULE. In the preceding pages we have treated with the common application of Simpson's first rule to ship calculations. Another method, equally, if not more simple, which is slowly gaining favor with naval architects is that devised by the Russian Tchibyscheff. This rule has the great advantage of employing fewer figures in its application ; more especially is this the case in dealing with stability calculations, and its usefulness in this respect is seen in the tabular arrangement given here. It has the additional advantage of employing a much less number of ordinates to obtain a slightly more accurate result and the use of a more simple arithmetical operation in its working out, viz. addition. As the ordinates, however, are not equidistant, it has the disadvantage of being inconvenient when used in conjunction with designing, and for this reason its use is advocated for the finished displacement sheet and calculations for G.Z. The rule is based on a similar assumption to Simpson's, but the ordinates are spaced so that addition mostly is employed to find the area. The number of ordinates which it is proposed to use having been selected, the subjoined Table gives the fractions of the half length of base at which they must be spaced, starting always from the half length. The ordinates are then measured off and summed, the addition being divided by the number of the ordinates, giving a mean ordinate, which multiplied by the length of base produces the area : — Sum of ordinates x Length of base = Area. |