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ship-section by the length of the load-water-line, and divide the load displacement by the product.
The coefficient of fineness of the displacement is equal to the product of the coefficient of fineness of the midship-section, multiplied by the mean coefficient of fineness of the water-lines.
The Tonnage of a ship, according as the word is qualified, may mean either the Displacement in Tons, the Burden, the Registered Tonnage, or the Tonnage by "Builders"" old measurement.
The Burden means the number of tons of lading which the ship is able to carry, in addition to the weight of her hull and equipments. It is obviously equal to the difference between the displacement when light, and the displacement when loaded; and, on the scale of displacement, supposing OH, Fig. 15, to represent the load draught of water, and oh the light draught, the burden is represented by the difference between the ordinates, HD and hd. Hence the burden of a ship, whose dimensions and figure, and light and load draughts are given, can always be calculated with precision. According to what has already been stated in Chapter II., the burden of a ship ranges from about one-half to two-thirds of her load-displacement, according to the heaviness or lightness of her construction. Wooden ships are heavier than iron ships of the same load displacement, and ships of war heavier than merchant ships. The following may be taken as ordinary proportions:
Per Cent. of Load Displacement.
Large ships, to be equally strong with small ships, must be made proportionally heavier; so that the weight of a large ship will form a greater percentage of the displacement, and that of a small ship a smaller, than the average stated above. This applies especially to the skin, the keel, and all longitudinal parts of the framing, whose weight should vary nearly as the displacement multiplied by the length, or, in similarly shaped vessels, as the displacement multiplied by its own cube root.
The total burden of a steam-vessel includes her engines and store of fuel; hence, to find her net burden, available for cargo, those weights must be subtracted from the total burden. The proportion which they bear to the displacement varies very much
in different cases, according to the speed, the figure of the ship, and the construction and efficiency of the engine; and it is likely to undergo very great diminution when improvements in design and in economy shall have been generally adopted in practice. The results of present practice, with moderately good design and economy, may be roughly approximated to by the following rules:
A steamer of 1000 tons' displacement, to go at a full speed of ten knots under steam, requires engines of the weight (including boilers) of about 125 tons, and a store of coal of one ton per hour of the voyage, or one-tenth of a ton per nautical mile.
The weight of engines, and of fuel consumed per hour, varies nearly as the square of the cube root of the displacement, and as the cube of the speed; but the weight of fuel for a given trip varies as the square of the speed.
[In algebraical symbols, let V denote the speed in knots per hour, and D the displacement in tons; then
Such calculations as these, however, give but a loose approximation; for the actual weight of engines and boilers, even according to ordinary examples, may range from four-fifths to once and a quarter of that given by the above rule, and the consumption of fuel within even wider limits (say from two-thirds to once and a half in ordinary cases, and from half to double in extreme cases), owing to the great differences in the economy of engines.
The burden of a ship may be computed approximately, by multiplying the area in square feet of the water-section midway between the load and light water-lines, by the difference between the load and light draught in feet, and dividing by 35 for tons. The area of that water-section may also be approximated to with considerable accuracy by a practised measurer, by measuring simply its extreme length and breadth, and multiplying their product by a coefficient of fineness, estimated by the eye at the commencement of this chapter; and this was the method of measuring the burden of ships introduced by Chapman into Sweden.
Combined Calculations of Buoyancy and Stability-Object of this Chapter-Arrangement of the Data--Arrangement of the Results of Calculations.
COMBINED CALCULATIONS OF BUOYANCY AND STABILITY.
THE object of the present section is to illustrate by an example, the manner in which the calculations of displacement, and of the positions of the centre of buoyancy and metacentre, can be conveniently combined in one tabular arrangement for practical purposes.
The methods of doing this are, of course, all identical in principle; but during the progress of naval architecture they have varied considerably in detail, and have been from time to time rendered more simple and concise. The arrangement adopted in this section is the most simple and concise yet known.*
The cross-sections are numbered from 1 to 17, commencing at the stern.
The ordinates or half-breadths at the intersections of the crosssections and water-sections, having been measured, are set down in the Table given at the end of this chapter.
The column on the extreme left of that Table contains the numbers of the cross-sections 1, 2, 3, 4, &c.
The next column contains Simpson's Multipliers, in their order, agreeably to the rules given in Chap. VIII.
Then follow the columns containing the ordinates,
Of these columns there are as many as there are water-sections; that is, in the present case, nine, including the base-line.
The columns containing ordinates are headed at the top with the numbers of the water-sections, and immediately below these with Simpson's Multipliers. The ordinates are ranged in as many lines as there are cross-sections; that is, in the present case, seventeen: being at whole-intervals apart.
Arrangement of Results of Calculation.—Immediately to the right of each ordinate is written, in differently-sized or differentlycolored figures, its product by the Simpson's multiplier proper to the line to which the ordinate belongs.
Immediately below each ordinate is written, in differently-sized
* This method was devised by the late Mr. John Wilson, chief draughtsman in the Surveyors' department of the English Admiralty.
or differently-colored figures, its product by the Simpson's multiplier proper, to the column to which the ordinate belongs. For example, at the intersection of the line belonging to the cross section 3 (for which the Simpson's multiplier is 2), and the column belonging to the water-section 3 W.L. (for which the Simpson's multiplier is 4), is the ordinate 9.87. Immediately to the right of that ordinate is written its product by the multiplier 2, viz., 19.74, and immediately below it is written its product by the multiplier 4, viz., 39.48.
The products written below the ordinates are added in lines; and the sum of each line of products is written in the column
headed "Half Areas÷ V.I." under the general heading "Vertical
Sections." The numbers in this column are proportional to the areas of the several vertical cross-sections; but to give the absolute values of those areas, they still require to be multiplied by 2, and by one-third of the vertical interval of the ordinates, (abbreviated into
Each of those numbers proportional to the areas of the crosssections is then multiplied by the proper Simpson's multiplier, found in the second column from the extreme left of the Table ; and the products are written in the column headed "Multiples of Areas." These multiples being added up, their sum (viz., 16563.20) is written at the foot of the column. It is then multi·V.I. 2.12 plied successively by one-third of the vertical interval( and by one-third of the horizontal interval (H.I._14.5) The
product (56572.52961) is one-half of the load displacement, in cubic feet, which being multiplied by 2 gives 113145.05922 cubic feet, the whole load displacement; and this being divided successively by 7 and by 5 gives 3232.71597, the Load Displacement in Tons.
Each of the numbers in the column headed "Multiples of Areas" is next multiplied by the proper "Multiplier for Leverage," contained in the column on its right. The multiplier for leverage for a given cross-section is the number of intervals by which that cross-section is distant from the first cross-section or commencement of the base-line.
The products are set down in the column headed "Moments; and having been added up, their sum (131032·80), at foot of
column) is multiplied by the horizontal interval (H.I.=14.5). The product (1899975-60) is not the absolute value of the moment of the displacement relatively to the first cross-section; but it bears. the same proportion to that moment which the sum of the column headed "Multiples of Areas" (16563,20) bears to the displacement. Dividing, therefore, that product by that sum, the quotient (1899975.60÷16563.20=114.7) is the horizontal distance in feet of the centre of buoyancy forward of the first cross-section.
Returning to the columns containing the ordinates, the products written immediately to the right of the ordinates are added in columns, and the sum of each column of products is written at the foot of the column, in the line marked "Half-Water-SecH. I. The numbers in this line are proportional to the 3 areas of the several water-sections; but to give the absolute values of those areas, they still require to be multiplied by 2, and by one-third of the horizontal interval between the ordinates (here
abbreviated into H.I.). Each of those numbers proportional to the areas of the water-sections is then multiplied by the proper Simpson's multiplier, as written in the line below it. The products are written in the next line again, marked "Multiples of Water-Sections," and being added together, their sum (16563·20) is written to their left. If the calculations have been correctly made, that sum ought to agree exactly with the sum of the column. headed "Multiples of Areas."
Each of the numbers in the line of "Multiples of Water-Sections" is next multiplied by the proper "Multiplier for Leverage," contained in the line immediately below. The multiplier for leverage for a given water-section is the number of intervals by which that water-section is below the load-water-section. The products are set down in the line marked "Moments;" and having been added together, their sum (57775-68) at the left end of the line, is mutiplied by the (V. I.=2·12). The product (122484-4416) is not the absolute value of the moment of the displacement relatively to the load-water-section; but it bears the same proportion to that moment which the sum of the line marked "Multiples of Water-Sections (16563-20) bears to the displacement. Dividing, therefore, that product by that sum, the quotient (122484-4416÷ 16563·2=7•39) is the depth, in feet, of the centre of buoyancy below the load-water-section.