1893.] On the Whirling and Vibration of Shafts. 3fi9 



The value of for wrought-iron or mild steel is 5389 0, and for 

 brass 37720. 



Having thus obtained formulae giving the whirling speed due to 

 each cause, on the assumption that all the others are neglected, it only 

 remains to find an empirical formula giving the resulting whirling 

 speed, when all the disturbing elements are taken into account, in 

 terms of the separately calculated whirling speeds due to the several 

 causes. Since the whirling speed in every case varies inversely as the 

 square root of the weight of the pulley (see Equation 4), the formula 

 for calculating the resulting speed was taken to be of the form 



................ (7) 



for two disturbing elements whose whirling speeds taken separately 

 are N\, N 2 ; or of 



NilTO/ y (Ny% 2 + N 3 2 Ni 2 + N.W) .......... (8) 



for three disturbing elements whose whirling speeds, taken separately, 

 are N\, N" 2 > Na. The formula may be extended to any number of 

 disturbing elements. It is not strictly accurate, for in addition to 

 the whirling speed varying inversely as the square root of the weight 

 of the pulley, it also varies as some function (0) of the distance of 

 the pulley from the nearest bearing and of the size of the pulley. 

 The experiments, however, justify to a remarkable degree the as- 

 sumptions that have been made in calculating the resulting whirling 

 speed. 



In calculating the speed at which a continuous shaft of given 

 diameter, supported on bearings placed at equal distances apart, and 

 loaded with pulleys on any or all of the spans, will whirl, the method 

 to adopt is to, first, find the span which will have the biggest whirl 

 (that is to say the span which carries the heaviest and most advan- 

 tageously-placed pulleys as regards whirling), and then to consider 

 this span and the spans immediately adjacent on either side. The 

 whirling speeds for the shaft and each of the pulleys on the three 

 spans have then to be calculated according to the rules laid down in 

 each case. The whirling speed for any pulleys on the two side spans 

 will, of course, be different according as that side span is an end or 

 an intermediate span in the line of shafting. Having found the 

 whirling speed due to each cause, the resulting whirling speed is 

 found from an equation of the same form as Equation (7) or (8) the 

 exact equation depending on the number of disturbing elements. 

 The speed thus obtained will be slightly less than the actual whirl- 

 ing speed. A nearer approximation to the actual speed might be 

 obtained by considering only those pulleys which lie near the centres, 

 or between the centres of the side spans and the bearings of the 

 middle span, neglecting the effect of those pulleys which lie beyond 



