36G Mr Stanley Dunkerley. [Nov. 23, 



The theory for the case of an unloaded shaft first received attention 

 at the hands of Professor Rankine,* who obtained numerical formulae 

 for the cases of a shaft resting freely on a bearing at each end, and 

 for an overhanging shaft fixed in direction at one end. 



The theory has been farther extended to the case of a shaft loaded 

 with pulleys by Professor Reynolds ; and the object of this investiga- 

 tion is to apply that theory and so obtain formula and by experi- 

 ment to verify them giving the critical speed in terms of the 

 diameter of the shaft, weights of pulleys, &c., in particular cases 

 applicable to the different conditions under which a shaft works. 



In many cases, as might naturally be expected, the " period of 

 whirl " of the shaft is merely its natural period of vibration. The 

 two periods are coincident in the case of an unloaded shaft (however 

 supported), and for a loaded shaft on which the pulleys are placed in 

 such positions that they rotate, when the shaft is whirling, in planes 

 perpendicular to the original alignment of the shaft. With pulleys 

 placed in any other positions, when the shaft is whirling, there is a 

 righting moment tending to straighten the shaft which does not exist 

 when it merely vibrates under the dead weight of the pulleys. 

 Hence, in an unloaded shaft, the period of whirl coincides with the 

 natural period of vibration; but, generally, in a loaded shaft, the 

 period of whirl is less than the natural period of vibration to an 

 extent depending on the size and positions of the pulleys. 



If, therefore, the period of disturbance (that is, the period of one 

 revolution) be decreased, the shaft runs true until that period ap- 

 proximates to the natural period of vibration of the shaft under the 

 given conditions. If the shaft now receive any displacement, how- 

 ever slight, a violent agitation is set up, which will be most marked 

 when the period of disturbance and the whirling period coincide. 

 As the period of disturbance is further decreased, the agitation be- 

 comes less and, at a period of disturbance slightly less than the 

 whirling period of the shaft, the shaft again runs true. 



As in the vibration of rods, so in the whirling of shafts, there are 

 a series of periods at which the shaft whirls. 



Investigation shows that the formulae obtained by considering the 

 combined effects of the shaft and only one pulley, or the combined 

 effects of two pulleys neglecting the effect of the shaft, are too com- 

 plicated even in the simplest cases for actual use. The only 

 alternative method is to consider the effects of the shaft and each of 

 the pulleys (whatever be their number, position, and size) separately, 

 and so obtain the whirling speed for each on the assumption that all 

 the others are neglected. By means of an empirical formula, the 

 whirling speed taking shaft and pulleys into account may be calcu- 

 lated from the separately calculated speeds of whirl. 



* Eankine's ' Machinery and Mill work,' p. 5-49. 



