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"time of rolling happen to be double, quadruple, or any even multiple of the periodic time of dipping, so that each roll "coincides with the rising part of the previously existing "dipping motion, the extent of the dipping motion may go on continually increasing to an amount limited only by the "resistance of the water." In short, when these ratios of the periods of dipping and rolling obtain, the ship is in a condition similar to that of a pendulum which receives periodically a fresh impulse at the end of its swing; and it is a matter of common observation how such an impulse, although in itself not of great magnitude, may by its repeated applications in the manner described lead to considerable oscillations. Dipping motions have not, however, the practical importance of rolling motions, and therefore they will not be further discussed. In vessels of ordinary form these motions are not nearly so extensive as in vessels of the Symondite type, and the reasons for the difference will be obvious.

1

Turning attention to the effect of fluid resistance upon the rolling of a ship in still water, that resistance may be subdivided into three parts:-(1) Frictional resistance due to the rubbing of the water against the immersed portions of the ship, and particularly experienced by the amidship parts where the form is more or less cylindrical. (2) Direct or head resistance, similar to that experienced by a flat board pushed through the water, and chiefly developed against the keel, bilge-keels, deadwood, and flat or nearly flat surfaces lying near the extremities of the ship. (3) Surface disturbance, which involves the creation of waves that move away from the ship, and have continually to be replaced by new-made waves, each creation involving, of course, a certain expenditure of energy, which must react upon the vessel, and be equivalent to a check upon her motion. The aggregate effect of these three parts of the fluid resistance displays itself in the gradual extinction of the oscillations when the ship rolls freely under the action of no external forces other than gravity and buoyancy; and if

observations have been made of the rate at which extinction proceeds in any ship, it is possible to infer from thence the total resistance for that ship, or for one identical with or very similar to her. But to estimate by direct calculation the value of the resistance for a ship of novel form, or for any ship independently of reference to rolling trials for similar ships, is not, in the present state of our knowledge, a trustworthy procedure. This difficulty in theoretical investigation arises chiefly from the doubtfulness surrounding any estimate of the "wave-making function" for an untried type. Having experimental data such as Mr. Froude has made available, it is possible to approximate to the first two parts of the resistance, but the third, as yet, seems outside calculation. For example, when the character of the bottom of a ship is known-whether she is iron-bottomed, or coppersheathed, or zinc-sheathed, and whether clean or dirty—it is possible to obtain the "coefficient of friction" for the known conditions; then knowing the area of the surface upon which friction operates, and the approximate speed with which the ship rolls, the total frictional resistance may be found within narrow limits of accuracy. Similarly, when the "coefficient of direct resistance" for the known speed has been determined by experiments on a board or plane surface, it may be applied to the total area of keel, bilgekeels, deadwood, &c., and so a good approximation made to the total "keel" or "direct" resistance. But the wavemaking function cannot be similarly treated, and so it becomes most important to make rolling experiments in still water, in order that the true value of the resistance may be deduced from the observations. The importance of the deductions arises from the fact that fluid resistance has very much to do with controlling the maximum range of oscillation of a ship rolling in a seaway. This will be explained in a future chapter; for the present it is sufficient to remark that, if the rate of extinction of still-water oscillations is rapid, it may be assumed that the range of rolling at sea will be greatly limited by the action of the resistance; whereas,

if the rate of extinction is slow, resistance will exercise comparatively little control over the behaviour of the ship at sea.

Rolling experiments in still water were recommended strongly by Bouguer in the Traité du Navire published in 1746, but their performance has only become common within the last few years. Mr. W. Froude, F.R.S., has conducted the greater number of those made on ships of the Royal Navy, and to him we owe our most valuable information on the subject; a few experiments have been made by officers in command. In the French navy such experiments have been made systematically for some years, and many of the results obtained have been collected and published. The objects of these experiments are twofold: (1) to ascertain the period of oscillation of the ship; (2) to obtain the rate of extinction of the oscillations, when the vessel is left free to move and gradually come to rest. Various means may be employed to produce the desired inclination, from which the vessel is to have her rolling motion observed. If she is small, she may be "hovedown," and, after reaching the required inclination, suddenly set free. But this is a process inapplicable to large ships, and the following is the plan usually adopted.

A number of men are made to run across the deck, from side to side, their motions being regulated by some concerted signal, so that they may run out from the middle line to the side and back again, while the ship performs a halfoscillation. By this simple means even the largest ships may be made to accumulate motion very quickly, and to roll through considerable angles, the running of the men being so timed as never to retard, but always to accelerate, the rolling. For example, her Majesty's ship Sultan was made to roll to an angle of 14 degrees from the upright by the motion of her own crew of about six hundred men, under Mr. Froude's direction; while the Devastation, weighing over 9000 tons, was made to reach a heel exceeding 7 degrees by four hundred men running eighteen times across her deck. If the motions of the men are not well timed, similar results

will not be obtained, and in some trials large angles of oscillation have not been secured, on account of non-compliance with this condition. When a sufficiently large range of oscillation has been obtained, the men are made to stand still, and the observations are commenced.

In order to determine the period for a single roll, careful note is taken of the times occupied by the ship in performing each of several successive single rolls; and in this way the fact has been established that vessels of ordinary form are practically isochronous in their rolling motions. Hence, in fixing the period for a ship, it is usual to observe how many oscillations (n, suppose) are made in a certain interval of time (T seconds, suppose); then the period

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Careful observations are also made of the extreme angles of heel reached at the end of each oscillation; the difference between the successive values marking the rate of extinction. A vessel starting from an inclination of (say) 10 degrees to port only reaches an extreme heel of 9 degrees to starboard, and then rolls back to 84 degrees to port; gradually coming to rest. These observations are commonly continued until the arc of oscillation has diminished to 2 or 3 degrees. Mr. Froude has devised beautiful automatic apparatus for recording the rolling motion of the ship in such a manner that the angle of inclination, at each instant of her motion, as well as her extreme angles of heel, can be traced, and the period also determined. But with the aid of the simplest apparatus it is possible to make all the observations needed, and in a future chapter the common plan will be described. The gradual degradation in the range of oscillation is represented by means of, what are termed, "curves of extinction"; examples of these curves, obtained from Mr. Froude's experiments, are given in Fig. 53, for her Majesty's ships Sultan, Inconstant, and Volage. A very brief explanation of the construction of these curves will suffice. On the base-line OX are set off equal spaces, each representing an oscillation;

and since each oscillation is performed in the same period, each of these spaces also represents a certain number of seconds. Any ordinate, drawn at right angles to OX, through the points marking these equal spaces, shows the extreme angle of heel reached at that particular oscillation; and the difference between any two ordinates so drawn shows the loss of range, or extinction of the rolling, in the corresponding number of oscillations. For example, after making

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twelve oscillations from the extreme angle (133 degrees) where the record of observations begins, the Sultan only reached an extreme angle of 8 degrees, the loss of range in that number of rolls being 5 degrees. Here the rate of extinction was slow, the vessel having a large moment of inertia, no keel, and only shallow bilge-keels, to assist the extremities in developing resistance to the motion. If there were deeper bilge-keels, the rate of extinction would be much more rapid.

Experiments have been made by Mr. Froude to show how

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