March 12, 1908J 



NATURE 



447 



THE USE OF GYROSTATS. 



A' 



T a recent meeting of the Physical Society a model 

 was exhibited which purported to illustrate Mr. 

 Brennan's mono-rail railway. Prof. Perry, president of 

 the society, made the following remarks, which he was 

 afterwards requested to edit and publish : — 



In 1874 two famous men made a great mistake in 

 endeavouring to prevent the saloon of a vessel from rolling 

 by using a rapidly rotating wheel. Mr. MacFarlane Gray 

 pointed out the mistake. It is only when the wheel is 

 allowed to precess that it can exercise a steadying effect ; 

 the torque or moment which it exerts is equal to the 

 angular speed of precession multiplied by the moment of 

 momentum of the spinning wheel. 



It is astonishing how many engineers who know the 

 laws of motion of translation are ignorant of angular 

 motion, and yet the analogies between the two sets of 

 laws are perfectly simple. I have set out these analogies 

 in my book on " Applied Mechanics." 



The last of these, betw-een centripetal force on a body 

 moving in a curved path and torque or couple on a body 

 rotating about an axis, is the simple key to all gyrostatic 

 and spinning-top calculations. When the spin of a top 

 is greatly reduced, it is necessary to remember that the 

 total moment of momentum is not about the spinning 

 axis (see my ". Applied Mechanics," p. 594)- Correction 

 for this is, I suppose, what introduces the complexity 

 which scares the students of the subject of 

 the vagaries of tops ; but in all cases that 

 are likely to come before an engineer, it 

 would be absurd to study such a correc- 

 tion, and consequently calculation is e.xcced- 

 •ngly simple. 



Inventors using gyrostats have succeeded 

 in doing the following things : — ■ 



(i) Keeping the platform of a quick- 

 firing gun level on board ship, however the 

 ship may roll or pitch. Keeping a sub- 

 marine vessel or flying machine with any 

 plane exactly horizontal or inclined in any 

 specified way. These were probably first 

 described by Mr. Brennan. It is easy to 

 effect such objects as these without the use 

 of a gyrostat. By means of spirit levels 

 it is possible to command powerful electric 

 or other motors to keep anything 

 always level. The actual methods employed 

 by Mr. Beauchamp Tower (an hydraulic 

 method) and by myself (an electric method) 

 depend upon the use of a gyrostat which 

 is really a pendulum, the spinning axis 

 being vertical. 



(2) Greatly reducing the rolling or pitching of a ship, 

 or the rolling of a saloon in the ship. This is the problem 

 which Mr. Schlick has solved with great success, at all 

 events in the case of torpedo-boats. 



(3) In Mr. Brennan's mono^rail railway, keephig the 

 resultant force due to weight, wind pressure, centrifugal 

 force, &c., exactly in line with the rail, so that however 

 the load on a waggon may alter in position, and although 

 the waggon may be going round a curve, the waggon is 

 quickly brought to a position such that there are no forces 

 tending to alter its angular position. The car leans over 

 toivards a sudden gust of wind or towards the centre of 

 curvature if going round a curved rail. 



(4) I need not refer to such matters as the use of gyro- 

 stats in the correction of compasses on board ship. 



Problems (2) and (3) are those to which I wish to refer. 

 It is to be remembered that without gyrostatic apparatus 

 a ship is necessarily stable, a mono-rail waggon is 

 unstable. 



Mr. Schlick uses a large wheel of ten or twenty tons 

 revolving about an axis EF (Fig. i), the mean position of 

 which is vertical. Its bearings are in a frame EFCD, 

 which can move about a thwartship axes CD. Its centre 

 of gravity is below this axis. Let the ship have rolled 

 through the small angle R from its upright position ; the 

 axis EF has precessed through the angle P from a vertical 

 position. Let $ stand for d/dt. Let the moment of 



momentum of the wheel about its axis be m. Now if the 

 ship were held fast so that she could not roll, we might 

 study the vibratory motion P. The effect of the roll is 

 merely to introduce a term inOR increasing P. Thus we 

 have 



I,6-P-f-yBP-weR + *P=o (i) 



where /rtP is a fluid friction introduced bv dash pots 

 acting at A and B, fcP is the righting moment of the 

 frame, and I, its moment of inertia about the thwartship 

 axis.- Now write out the usual equation of motion of the 

 ship vibrating about a longitudinal axis through its centre 

 of gravity, its moment of inertia being I, but introduce a 

 moment ni6P tending to diminish R. 

 Then we have 



le-R + FeR4-weP-l-a(R-a) = o. ... (2) 

 if a = a„sin(j( is the thwartship inclination of the sea to 

 the horizontal, and a is the righting moment of the ship 

 per unit angle, being the weight of the ship multiplied by 

 the metacentric height. F9R is the moment due to friction 

 against the sea. 



Solving these equations just as if 6 were a constant, we 

 have from (i) 



f«flR 



so that (2) becomes 



~liff-+je + 6' 



ie-+Fe + a + — 



i^ff'+fe+ij 



(3> 



Clearing of fractions we find 

 {Uie^ + (FI,+/I)ff^ + {al^ + />l + »fi + F/)ff' + (iF + a/)e + a/>'iR = 

 {I^e'+/e + i)aa. ... (4) 

 Replacing $- by —5-, and 9' by <j' (see my " Calculus 



R 



for Engineers," p. 237), we can at once express — if 



R„ is the amplitude of the roll, and, of course, 

 r, '"'& 



(5) 



NO. 2002, VOL. yy] 



I am here studying the forced vibrations, and not the 

 natural vibrations. In any particular case it is quite easy 

 to calculate Rj/oj for a number of values of g, and in- 

 formation is obtainable which is quite different from 

 what comes from a study of the natural vibrations (that 

 is, taking = 0). Besides, it is the very easiest kind of 

 arithmetical calculation, replacing rather troublesome 

 mathematics. To many, indeed I may say to all students, 

 the calculation of the unreal roots of a biquadratic is 

 troublesome, and this must be done if the natural vibration 

 is to be studied. It is obvious that the real parts of the 

 roots of the resulting equation in R (when a is o) are 

 negative, and therefore the motion is stable.' 



If, however, we make a of (2) negative, as it is in the 

 1 The well known condinons ihat the re.il parts of all the rrots of 0i+a$'< 

 +ie'^+ce+i/=o shall be negative, are that a, I', c, and d shall be positive, and 

 also that adc' c^~ a~d shall be positive. 



