;54 



NATURE 



[September 14, 191 1 



roll and time, it was easy to determine the law which 

 represented the resistance in terms of the angular velocity. 



In applying this to waves, all that i- necessary is to take 



nt of the fact that the position of equilibrium about 



which the ship oscillates is the normal to the effective 



wave-slope. This normal has a definite oscillation about a 



fixed vertical. It is, therefore, possible to determine the 



of inclination in terms of time. 



\^ these angles of roll may be considerable, the assump- 

 tion upon which the general solutions for unresisted 

 rolling, already given, were based will not hold. The actual 

 en'ts of stability depend upon the form of the ship and 

 the position of its 'centre of gravity, and as these vary in 

 different ships it is onlv possible to obtain the relation 

 between inclination and 'time bv a special investigation in 

 each case. A solution bv a method of graphic integration 

 was devised by Mr. W. 'Froude and has been applied to a 

 very small number of cases. The information necessary to 

 obtain a solution in any one case is as follows : — 



(0 A curve of righting levers in terms of angle of 

 inclination. This is called a curve of statical stability. _ 



1 2 ) The' form and period of the wave on which the ship 

 is supposed to be placed broadside on. 



(3) The constants which determine the actual value of 

 the resistance moment in terms of the angular velocity. 

 These can be obtained by rolling the ship in still water 

 and observing the rate of extinction of rolling when that 

 extinction is due to resistance only. The form of the curve 

 of extinction can be obtained by rolling a model of the 

 ship, but the actual ordinates of the curve for an actual 

 ship can only be obtained by experiment on the ship 

 herself, or by inference from a similar ship of approximately 

 the same size, form, and arrangements. 



A consideration of these three necessities for the solution 

 of one particular case shows that a considerable amount 

 of work is necessary for determining the angle of inclina- 

 tion in terms of time. Even this solution can only be 

 made for one assumed maximum angle of inclination as a 

 starting condition. For instance, in any case where a 

 ship is assumed to start with a maximum inclination of 

 20° it is only possible to obtain one solution of angles of 

 inclination in terms of time. If we take another maximum 

 angle of inclination, another complete solution is necessary. 

 The work of each solution is considerable. 



For ships which vary much in draught and condition of 

 loading it is evident that for each ship the work of com- 

 plete investigation for all the conditions of loading of 

 different waves and different angles of maximum inclination 

 is very great. For this reason .the investigation of rolling 

 by the Froude graphic method has only been made for a 

 very small number of cases, and our knowledge of the 

 actiial angles of inclination of ships obtained by this 

 method is verv small. 



The curve of statical stability is worked out for many 

 ships in a few conditions of draught and position of centre 

 of gravity. These curves are of little practical _ value, 

 because they only serve as comparisons between ships. _ It 

 is assumed' that" if a ship has a fair range of statical 

 stability, i.e., that the angle of vanishing statical stability 

 is not less than, say, 60°, and the maximum righting lever 

 is not unlike some' previous ship which " has been to sea 

 and come home again " safely, this ship will be safe. This 

 assumption is based on the belief that only what has 

 happened to previous ships will happen to the one in 

 question ; that is, thai the contingencies of waves will be 

 the same in all cases. Rut when we find that occasionally 

 ships are missing we air- compelled to ask ourselves the 

 ion— is il possible that some occasional contingencies 

 of sea or ship, or both, may exist which will produce a 

 dangerous and perhaps fatal roll? 



Mr. W. Froude's investigations were made for a uniform 

 11 of waves. He showed that in unresisted rolling a 

 ship initially at rest and in the upright position is 



T p 

 upon by a uniform series of waves such that 



where p and q are the smallest whole numbers which 



express this ratio, then the rolling of the ship will be in 



mum inclination in each roll gradually 



asing, and again gradually diminishing, and Si 

 The period of occurrence of the maximum of maxima will 



T. The number of times the ship passes throug 



NO. 2185, VOL. 87] 



upright in each complete cycle is 2p or 29, whichever is- 

 the smaller. The ship is upright at the middle of the 

 cycle, and on either side of this middle there is an equal 



maximum which is approximately 9, — - , and never 



exceeds this value (where 8, is maximum wave-slope). If 

 T is much larger than T„ and therefore p is much 



larger than o, then the value of 8, '' approaches 0/ 



and is less than the wave-slope. If T is much smaller 



than T,, then the value of 8, — '— approaches 8,. If T is 



nearly equal to T,, then 0, -— approaches a high value. 



From this it is seen that our investigations in unresisted 

 rolling may be over a very wide field, but would produce 

 no definite result in the matter of finding cases of large 

 angles of roll in practice. We can only obtain valuable 

 results when resistance is included. 



Mr. R. E. Froude in 1896 was led to deal with the 

 subject of non-uniform rolling of ships in an assumed 

 uniform system of waves which did not synchronise with 

 the ship, such as is dealt with above for unresisted rolling, 

 and he dealt with the effect of resistance in such a 1 

 He pointed out that there is a particular phase-relation 

 between the ship and the wave which will produce uniform 

 rolling, just as there is in the case of a synchronous system 

 of waves. If at any stage for any reason the rolling is of 

 the cyclic character considered in non-resisted rolling, then 

 the resistance must gradually introduce uniformity, because 

 the rolling is made up of two seas of oscillations — 



(1) That due to the rolling relatively to the water-surface, 

 such as would occur in undisturbed water. 



(2) That due to the oscillation of the water-surface itself. 

 caused by the passage of the wave. 



We have already seen that the resisted oscillation in 

 undisturbed water gradually decreases when the vessel is 

 left free to oscillate, but takes place in practically uniform 

 time T. The oscillation of the water-surface is forced on 

 the ship and causes a simple harmonic oscillation of the 

 ship in time T,, in algebraic addition to that due to the 

 free resisted oscillation. When the maximum angle of a 

 roll due to the free oscillation coincides with the maximum 

 angle due to the forced oscillation of the wave, we shall 

 have a maximum extreme inclination which is the sum of 

 that due to the free and the forced. When they are in 

 opposition we shall have a minimum extreme oscillation 

 which is the difference of these two. At stages betweei 

 coincidence and opposition we shall have extreme at 

 varying between maximum and minimum. As time goes 

 on the extreme angle due to the free oscillation gradually 

 decreases under resistance, and the sum and the difference 

 referred to above approximate to each other, and the rolling 

 becomes that due to the wave alone. We have seen that 

 in the case of unresisted rolling where the wave and the 

 ship synchronise there is an addition to the angle of 

 inclination for each passage of the wave, and were i 

 for resistance these accumulated increases would cause the 

 vessel to upset. But in the case of resisted rolling 

 increase of extreme angle of roll causes an increase in the 

 work done upon the resistance of the ship, and when the 

 increase in work done in increasing the angle of heel 

 by each passage of the wave equals the work done in 

 increasing the resistance incurred in swinging through this 

 1 angle, then we shall have a balance of condition 

 and a uniform angle of roll. The angle at which this 

 balance takes place depends on the period and maximum 

 slope of the wave and the coefficients of resistance between 

 the ship and the water. For instance, with a maximum 

 wave-slope of 3 and with a ratio of ship to wave-period 

 of 1.1 the value of the angle of ultimate uniform rolling 

 in the rase of II. M.S. Revenge was found to be 130" 

 without bilge-keels and io-8 c with them. In the case of 

 synchronism of the ship and the wave, the rollii 

 uniform alwavs and reaches a maximum of 41-1° without 

 and 14-85° with bilge-keels. The nearer the wave and 

 ship are to synchronism, the larger is maximum it 

 tion reached before uniform rolling sets in and during 

 uniform rolling! Resistance is of much more importance 

 in the 1 of si chronism. If the ratio of ship to wave- 



period be 1.3, the maximum angle before uniform rolling 



