368 Mr. Stanley Dunkerley. [Nov. 23, 



in which ?, 0, E, I have the same meanings as in Equation (1), and 



W = weight of pulley in Ibs. ; 



c = distance of pulley from nearest bearing in feet ; and 

 = some numerical coefficient depending not only upon the 

 manner in which the shaft is supported, but also upon 

 the position of the pulley on the span which carries it 

 and the size of the pulley. 



For any particular mode of support of the shaft the coefficient 6 is 

 some function of the ratios c/l and c/&, where I = length of span 

 which carries the pulley, and k = V g (A B)/W, where A and B 

 are the moments of inertia of the pulley about the axis of the shaft 

 and a diameter of the pulley through its centre of gravity respec- 

 tively, both being expressed in gravitation units. Assuming, there- 

 fore, certain values for c/l and c/k, tables have been drawn up giving 

 the corresponding values of 9. For each value of c/l there are two 

 limiting values of c/&, viz., infinity and zero the corresponding values 

 of k being zero and infinity. When k = the pulley may be con- 

 sidered simply as a dead weight, so that the "inferior period of 

 whirl," as it has been termed, is the natural period of vibration of 

 the light shaft under the given conditions. The " superior period of 

 whirl," that is to say, the whirling period when k = oo^, is the inferior 

 period multiplied by some function of c//, and assuming the shaft to 

 whirl at a speed corresponding to the superior or inferior limit, it 

 would do so in such a manner that the pulley still rotated in a plane 

 perpendicular to the original alignment of the shaft. These limiting 

 values of the speed, below and above which whirling is impossible, 

 have been calculated in each case. 



Investigation shows that in a continuous light shaft supported on 

 bearings placed at equal distances apart, the increase in the whirling- 

 speed due to those spans which are not immediately adjacent to the 

 loaded one, on either side, cannot exceed by above 2 or 3 per cent, 

 the whirling speed when only the loaded span and the spans im- 

 mediately adjacent to it are taken into account. In other words, the 

 stiffening effects of only those spans immediately adjacent to the 

 loaded one, on either side, need be taken into account in calculating 

 the whirling speed for the shaft. 



By substituting the proper values for the constants in Equation (4) 

 we obtain the equations 



N = d 2 /\/Wc 3 , for a solid shaft (5), 



and = v/(di 4 -d 2 4 )/ -/ We 3 , for a hollow shaft (6), 



where W, d, c?i, d 2 have the same values as before, and is some 

 numerical coefficient depending on the value of and of the con- 

 stants in Equation (4) . 



