Whirling of an Excentrically -loaded Overhung Shaft. 515 



by showing that when the second integrations are carried 

 out over the appropriate ranges and the integrals added, we 

 recover \. 



It may be remarked that the latter results may be applied 

 to the complete circle by making a = 2ir (fig. 7). The second 

 range for r then disappears, and for the whole range now 

 -extending for all values of 6 from r = to r = 2 we get 



Z7W=7? (40) 



which needs to be doubled in order to take account of 

 negative values of 6. 



This completes the investigation for an arbitrary a (less 

 than 27r), when n = 2. Since even for the complete circle 

 (a =2tt) the case n = 3 leads to elliptic integrals, there is no 

 -encouragement to try an extension to other values of a. 



Telling Place, With am. 

 March 31, 1919. 



XLV1I. The Whirling of an Excentrically -loaded Overhung 

 Shaft. By S. Lees, M.A. } ('ollege of Technology, Man- 

 chester *. 



§ 1. rj^HE general problems arising in connexion with 

 JL the whirling of shafts have been treated at some 

 length by several investigators, prominent amongst whom 

 may be mentioned Greenhill f , Dunkerley J, and Chree §. 

 The latter, in particular, has given a very elaborate treatment 

 of the subject on mathematical lines, but does not give any 

 •attention to the practical problem of the whirling of shafts 

 with excentric loads. 



§2. In what follows, an attempt is made to deduce the 

 dynamical equations giving the motion of such an overhung, 

 exeentricallv-loaHed shaft. The writer has taken into account 

 the influence of the rotational inertia of the whirling load, but 

 has made what seems a reasonable assumption, namely, that 

 the whirling load can be taken as a perfectly true fly wheel, 

 mounted slightly excentrically. This simplifies the mathe- 

 matical discussion somewhat. It is also further assumed 

 >that the weight of the shaft can be neglected. 



* Communicated bv the Author, 

 t Greenhill, Proc. Inst. Mech. Eng. 1883. 

 t Dunkerley, Phil. Trans. Roy. Soc. A, 1894. 

 § Chree, Phil. Mag. May 1904. 



2N2 



