1893.J On the Whirling and Vibration of Shafts. 367 



Unloaded Shaft. In the case of an unloaded shaft, the whirling 

 speed is given by an equation of the form 



a ................. (1), 



in which 



w = weight of shaft in Ibs. per foot run, 

 g = gravity, 



E = Young's Modulus (in Ibs. per sq. ft.), 

 I = geometrical moment of inertia of cross section of shaft 



about a diameter, 



w = angular velocity of shaft when whirling, 

 I = length in feet of one of the spans, and 

 a = some numerical coefficient depending on the manner in 

 which the shaft is supported. 



By substituting the proper value for the coefficients in Equation (1), 

 we obtain the equations 



N = b d/P, for a solid shaft ................. (2), 



and = &-/(di 8 + rf 2 2 )/Z 2 , for a hollow shaft ...... (3). 



In these equations 



N = number of revolutions shaft makes, per minute, when 



whirling, 



d = diameter of the solid shaft in inches, 

 di 9 dz the external and internal diameters of the hollow shaft, 



both in inches, and 

 6 = some numerical coefficient depending oil a, and also upon 



the constants in Equation (I). 



For wrought iron or mild steel, b = 3331 or and for brass, 2222 dr. 



It should be noticed that each span of a continuous shaft sup- 

 ported on bearings placed at equal distances apart whirls indepen- 

 dently of the rest. 



Loaded Shafts. The cases for a loaded shaft as also those for an 

 unloaded one which have been considered in the investigation, em- 

 brace all those which are likely to arise in practice. With a con- 

 tinuous shaft supported on three or more bearings and loaded with a 

 pulley on one of the spans, general equations have been obtained for 

 unequal spans, but the complete investigation has only been 

 extended to the case of equal spans a relation between the spans 

 which almost invariably exists in actual shafting. The results of 

 the investigation show that however the shaft be supported, the 

 whirling speed due to a single pulley on a light shaft is given by an 

 equation of the form 



= v/foEI/Wc 3 ) ... ............. . (4), 



2 c 2 



