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are R1, R2, Rz . . ., then for speed VDV1, VDV2, VDV: of the ship, the resistance will be D'R1, D3R2, D2Rg.
To the speeds of model and ship thus related, he applied the term “corresponding speeds." This law expresses the resistance due to surface friction, plus wavemaking resistance, the former being commonly referred to as skin resistance and the other as residuary resistance, embracing as it does, the resistance caused
by the motion of the waves and the drag of dead water eddies, such as are formed at abrupt endings to bossings, the siding of stern posts and in the wake of propeller struts. The skin resistance is proportional to the area of wetted surface, and is responsible for almost the total resistance up to about 8 knots speed. Beyond this speed the total resistance increases rapidly, showing the effect of the residuary resistance. This will be more readily understood, when we recollect that the wave undulations progressively increase in height with increases in speed, and that the crests of these waves are accountable for about 95 per cent of the total residuary resistance, the remaining 5 per cent, as already stated, being due to eddies, etc. Referring to the diagram here reproduced, showing curves of residuary and skin resistances, “the graduated undulations in the residuary resistance curve are due to quasihydrostatic pressure against the after-body, corresponding with the variations in its position with reference to the phases of the train of waves comprising the wave line profile, there being a comparative excess of pressure (causing a forward force or diminution
of resistance) when the after-body is opposite a crest, and the reverse when it is opposite a trough. Their spacing is uniform at a uniform speed, because waves of given speed have always the same length; it is more open at the higher speeds, because waves are longer the higher their speed; their amplitude is greater at
the higher speeds, because the waves made by the ship are higher; and their amplitude diminishes with increased length of middle body, because the wave system by diffusing itself transversely loses its height.”
Froude found that, at the lower speeds, two ships, one 200 ft. and the other 240 ft. in length, had the same residuary resistance; the difference in the larger vessel was simply due to its increase of skin friction due to the greater wetted surface. At 13.15 knots, however, the 240-foot vessel had the lesser total resistance of the two, owing to her position on the residuary resistance curve coming in a hollow; the consequent diminution in this resistance was greater than her increase of skin friction.
The resistance depends on the relative placing of the after-body and the wave system, and the length spacing of the wave system depends on the speed, therefore the position of after-bodies, which is specially favorable at some given speed, may be specially unfavorable at a higher speed, and at a higher speed still may be favorable again.
This it is which explains the economy with which some vessels attain certain speed whilst others of almost identical form, but slight variation in length, fall short of the others' performance.
To apply the investigations of Froude to actual ships, it is usual to make a model of the proposed ship and run it in a tank, and from the data obtained apply the law of comparison. For example, if a model be made of a liner 700 feet long on a scale of
inch to the foot, and the required speed of the ship be 24 knots, at what speed will the model require to be run to correspond with the desired velocity ? “In comparing similar ships, or ships with models, the speed must be proportional to the square root of their linear dimensions." Therefore the model will be
= 871 inches, or 7 feet 34 inches, and the ratio of linear dimensions,
7.29 and speed corresponding to 24 knots,
24 : V96 = 2.45 knots. In like manner,
if we are working from the known speed of another ship, say, of 600 feet length, then:
700=1.16 ratio of linear dimensions, and
24 : V1.16=25.8 knots, corresponding speed of the 600-foot boat.
APPLICATION OF FROUDE'S LAW. It is, however, in dealing with data derived from trial performances that the law of comparison is invaluable to those having the responsibility of powering ships. For, given the trial data of the ships, we may apply this to other vessels of similar form to obtain the I.H.P. necessary to drive them at a stated speed. Of course, we assume that the efficiency of the engines, boilers and propellers are equal in both cases, otherwise that their coefficients of efficiency are the same. So that when we know the displacement, power, and speed of a given ship represented by D, P, and V, and it is required to estimate the I.H.P. from a proposed vessel of like form of D1, P1, and V1, then,
Di& (1) Pi=
(2) Vi= 'V. Substituting values,
X 22.1 17,878
= 24.4 knots. We may also run a speed curve of the known vessel, where progressive runs have been made, as shown in Fig. 53, and from this deduce the proposed vessel's corresponding curve with the aid of the formula given.
The curve illustrated is that of a 56-ft. vedette pinnace, and it is proposed to deduce the power curve of a 21 knot speed launch from it, being a type of similar form. Displacement of vedette.
13.75 tons. Displacement of speed launch 22.50 tons. The corresponding length L1 of the speed launch would be obtained from the length of the vedette and the ratio of the displacements.
x 56 feet = 66 feet.
13.75 Corresponding speed,
X 19.25= 20.85 knots.
13.75 Corresponding power,
X 315 = 558 I.H.P.
13.75 So that after the derived curve has been plotted from the spots calculated as above for various speeds, it must be continued in the same contour until it is opposite the 21-knot ordinate, when the required power may be read off.
STANDARD CURVES OF POWERS.
Taylor in his “Resistance of Ships" advocates the adoption of a "standard” displacement in applying the Law of Comparison, to which all trial particulars should be reduced, and for this purpose takes 10,000 tons as a basis, giving tables of factors to facilitate the reduction of the speed and power data possessed, to this standard displacement.
He makes each curve cover a range of one knot, after the manner shown on Fig. 54. As an example of the method employed in estimating the indicated horse power by the aid of these standard curves and tables, let us postulate that the power is required for a proposed ship of: Length
440 feet. Breadth
48 feet. Draught
19.5 feet. Displacement
7,000 tons. Coefficient, s
181 knots. Then to reduce 10,000 tons displacement, dimension, speed, and power factors are calculated.