448 



NA TURE 



[March 12, 1908 



Brennan rase, i^ will be found that the motion is not 

 stable. iCven without friction the vibration would become 

 great.'r and greater, and friction makes matters worse. 

 Indeed, no form of the Schlick method can be applied to 

 the Brennan waggon. But to return to the ship. 



It will- be found that the amplitude of P is much 

 ■greater than that of R, and in practice it is necessary to 

 have stops to prevent P becoming too great. Of course, 

 when further increase of P is prevented by a stop, the 

 roll proceeds as if the wheel were not spinning. 



I have not seen it mentioned, but I should think that 

 Mr. Schlick would let his wheel revolve like Mr. Brennan 's, 

 in a verv perfect vacuum inside a case, because the power 

 wasted in friction of a wheel against an atmosphere is 

 proportional to the density of the atmosphere. I have 

 found that the best shape of wheel is one like a fly-wheel 

 with a thin disc inside the rim instead of arms ; there is 

 more moment of momentum per pound to be obtained in 

 this wav than by building up a wheel like a compound 

 disc, as a gun is built up of tubes shrunk on ; and also 

 it is much better than the form of wheel adopted by 

 I. aval in his turbine. I need not say, also, that the 

 moment of momentum per pound of steel is proportional to 

 the radius of the wheel ; the greater the radius, therefore, 

 the better. 



It is assumed that by the use of bilge keels and rolling 

 chambers, and as low a metacentric height as is allow- 

 able, we have alreadv lengthened the time of vibration 

 and damped the roll R as much as possible. Using (2) 

 or (5), we find P if R is known, and usually the quick 

 vibration is much more magnified in P than the slow one. 



Let us consider a numerical example. Using engineers' 

 units, let I = io*, = 2x10', I, = 7000, !) = 70,000, 

 m = 2-5Xio°, and F = 4Xio'', and let us find answers to 

 (4) for two values of /. These nuj»Tbers are nearly right 

 for a vessel of 6000 tons, metacentric height 18 inches, 

 with a natural period of fourteen seconds. Its gyrostat 

 wheel weighs about ten tons, with a 6-feet radius and an 

 angular velocitv of 100 radians per second. The frame 

 and wheel have an oscillating period about the axis CD 

 of about two seconds. These answers are compared with 

 the case of the gyrostat not in action, that is, )n = o, or 

 precession prevented. 



If the gyrostat is not in action, it is easv to see that 



„q^. ... (6) 

 damping term 2~ '^ with a 



The natural vibration has : 



periodic time nearly 2jr . / -. 

 A' a 



I take F such that the amplitude diminishes by about 

 25 per cent, in one period (about fourteen seconds). It 

 will be noticed that the F term of the formula (6) is 

 important only near the critical q or 17 = 0-4472. It will 

 be found that the F term in (4) is of insignificant effect. 



Rn 



;\gain, the values of - get large for large values of 



0, because there is a quick natural vibration as well as a 

 slow one. I have not thought it worth while to tabulate 

 these higher values. 



NO. 2002, VOL. 77] 



It is interesting to calculate P„-^a„ for all values of 

 q, and especially for the larger values of ij. 



Free Vibration. — Using the above numbers and = 0, 30 

 that the ship is gradually coming to rest, we are led to 



R = Ae"''-''-^' sin 0-447/ ''-' 



if the gyrostat is not acting. This is a periodic lime of 

 fourteen seconds, and the damping is such as to reduce 

 the amplitude of roll by 25 per cent, in each complete 

 period. 



When the gvrostat is acting and 7 = 50, ooo, we are led 

 to 



R = Ae-»*--'" sin o-324/+Be-:'-™'!in (2-519/-I-C) . (8) 

 We may neglect the quick vibrations of 2^ seconds' period, 

 which are damped out very rapidly. The slower have a 

 period of nineteen seconds, the amplitude of roll being 

 diminished by 30 per cent, in every complete period. Note 

 that P„ = 5o R„ if 5 = 2.519, and P„=ii.3R„ if ((=0:324. 



When the gyrostat is acting and /=3Xio^, or six limes 

 as great, we are led to 



R = A€-»-»2' + Be-''^-2' + Cf-»-2''^',<;in 1703/, . (9) 

 so that the slower periodic motion has disappeared, and 

 the quick one, the period of which is nearh" 3-7 seconds, is 

 rapidly destroyed. For both (8) and (q) it is interesting 

 and easy to calculate P. 



In solving the biquadratics which lead to such answ-ers. 



o roots 

 'I being 

 the last 



(lol 



let it be noticed that we are led usually 



— a + l3i and —in±iu, where «=v'— i, " and 



much smaller Ih.-in a and p. If we leave ou 

 two terms of 



e^ + ne^ + A0- + ,-e + /i=o . . . 



we get the larger roots, approximately ; if we leave out 

 the first two terms we may not get ni, but we get a good 

 approximation to ii, and it is n which it is most important 

 to know. The following is a quick method of finding the 

 roots with any amount of accuracy that is required. We 

 know that ' 



c = 2(a ■>■„,), 



* = a- + 6- + m- + It" +4am, 



( = 2w[a- 4- ;8-) + 2a{in- + ir), 



The numerical example given above, where f = 5Xio'', 

 requires us to solve 



e' + 7-l6e-'-F 19 42fl--F 1-836 + 2 = 0. 

 First assume that t» = o, so that a = 3.5S. We see then 

 that the sum of a'-l-^- and m' + n' is 19-42, and their 

 product is 2, so that we can find them. 

 .V---I- ig-42.r + 2 = o 

 give^ d- + S^= 19-42, w- + »-=o-i03. 

 Then 0-915 or |r-= 19-42OT + 3-58 x 0-103 

 or ju = 0-0282. 

 Secondly assume that w = o-02S2, so that o = 3-5i;i9; taking 

 .r- + 1 9 -02.r -I- 2 = o we cget a- + 0- = IQ'Ol ; 7«'- -F «- = o-Io5I5, 

 Af = 0-915 = i9-02/« + 3-553 X 0-105 gives OT =0-0285. 



.'\ssuming m to have this value, we may proceed to a 

 third calculation. In this way w-e get closer and closer 

 to the true value of m, and therefore to the true values 

 of o, j3, and n. In practice I find that the two calcula- 

 tions such as I give here are sufficient. 



It may be taken as roughly true from (4) that the 

 effective moment of inertia of the ship is increased from 



I to I + -,„, so that the time of a slow vibration is multi- 



0- 

 plied by 



(i + .H=/Ih)*- 

 If all ships and their gear are similar, il will hi- found 

 that iii-/Ib is inversely proportional to the dimensions. 

 Thus if a lOo-ton bo.at has its period increased h\ 50 per 

 ccnt., then a perfectly similar ship of 2700 tons will have 

 its period lengthened by only 19 per cent. 



It may be, however, that the proportions should be 

 different in vessels of different size, and it is not fair 

 without further experience to make a coiriparison which 

 seems so unfavourable to the method. Besidi-s, experi- 

 ence alone can show how the dash-pot friction may depend 



