In the figure we have the centre of gravity G to G1, 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

The new water line is at W1L1 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 1 and LOL1, are of equal angle, so that

GG1 WW1 LL1 WW1+ LL1.

==

=

GM WO LO

=

1

WO+LO

But WW1+ LL1 is the change of trim, and WO+LO is the length of the vessel = L, then

Calling this change of trim 24 inches, and assuming that the point of intersection "O" 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 P×1=M1 the moment to change trim, and

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:

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: —

M" =

832.142

12

X .0001725

9.95 foot-tons,

as against 9.84 foot-tons found by actual calculations, a difference too insignificant to affect noticeably the change in trim.

This moment 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."

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 :

P

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 equal to the distance between WL and W1L1 whose depth is " Let the weight be now moved to its definite position at a distance I forward of C. G., then

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. of the original water plane, it is obvious that the

new line will have its centre of flotation somewhat further aft, and consequently the tangent of the angle W1OW2 will be less than that of L1OL2. 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 "O" to stem 60 feet and "O" to post 40 feet, the water line length being 100 feet, we have:

60

Immersion forward 6 × 10′′=6 inches | Total change
Emersion aft
× 10 4 inches ( 10 inches.

100

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

No. of ordinates

× Length of base = Area.

The employment of this rule to find the volume of displacement and the other elements usually tabulated on the displace ment sheet is shown on the attached Tables. The number of stations used is ten, as in the case of Simpson's rule, but for clearness the after body five are indicated by Roman numerals, and the fore body ones in Arabic. The displacement length is 600 feet, therefore by taking the fractions given in the preceding table for ten ordinates and multiplying them by 300, we shall obtain the distance of the displacement sections apart. These distances from the half-length and the sections are here given as used for the Table, but it will be observed that the water lines are spaced to suit Simpson's first rule for the vertical sections as no advantage would be gained by the use of Tchibyscheff in this direction, owing to the fewer number of water lines generally necessary. The various operations in the Table will be clearly understood from the headlines of the respective columns.

As already pointed out, the great value of this rule is in the calculations to obtain cross curves of stability, specimen tables of which are also given. The fewness of the sections necessary, and the fact that the integrator saves the calculator the tedium of adding up, tells greatly in favor of the adoption of this rule for these calculations both as a time saver and an eliminator of the chances of error.