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XXVII, The Lateral Vibration of Loaded Shafts in the Neigh- 

 bourhood of a Whirling Speed. — The Effect of Want of 

 Balance. By H. H. Jeffcott *. 



1. "TTTHEN a shaft whose centre of mass does not lie 

 T 1 on its axis of figure (the shaft then being said 

 to be " out of balance, "), is rotated, its geometrical axis ceases 

 to remain straight and in coincidence with the axis of the 

 bearings, but becomes bent and rotates round the latter axis. 

 The amount of this bending depends on, among other things, 

 the relation of the speed of rotation to that of any of the 

 free lateral vibrations of the shaft. So long as the speed of 

 rotation is not too ne;ir that of any of these vibrations the 

 shaft remains nearly straight unless it is badly "out of 

 balance. " If, however, the speed of rotation increases so as 

 to approach that of any transverse vibration the bending 

 becomes noticeable, — sooner or later according to the per- 

 fection of balance of the shaft — and increases rapidly ; and 

 the shaft bends in one or more loops according to the 

 particular transverse vibration whose speed is being ap- 

 proached. 



If the speed is maintained constant at such a value the 

 bending may become excessive and is ultimately restrained 

 by the action of the bearings. And if the speed changes 

 from one slightly below to one slightly above that of the 

 transverse vibration the character of the motion changes. 



In particular, for the slowest transverse vibration ox a 

 shaft put out of balance by a single mass, the motion changes 

 from one in which the centre of mass is nearer to the axis 

 of the bearings than is the elastic centre to one in which the 

 reverse is true. In this change the shaft appears to shiver. 



Well above the speed of the transverse vibration the shaft 

 settles down again to more or less steady running with the 

 axis nearly straight until the speed approaches that of 

 another transverse vibration. 



When the shaft is considerably bent it is said to " whirl." 

 The speeds of transverse vibration are spoken of as the 

 " whirling speeds." 



In the foregoing when the shaft is horizontal, the straight 

 form, or the axis of the bearings, is to be understood to refer 

 to the gravitationally deflected form. 



It is proposed in this note to discuss to some extent how 

 want of balance causes these phenomena, and to what extent 

 in good practice balancing should be carried. It should be 



* Communicated by the Author. 



