516 



Mr. S. Lees on the Whirling of an 



To specify the position of the system of shaft and flywheel,, 

 three variables will be required. These will be taken as 

 (i.) the deflexion s of the free end of the shaft from the 

 un deflected position j (ii.J the angular slope 6 which the 

 deflected, axis of the shaft at the free end makes with the 

 axis at the constrained end ; (iii.) the angle $ which the line 

 joining the O.Gr. of the flywheel to the centre of: the shaft 

 .makes with the line through this centre, perpendicular to 

 the shaft and in the plane of bending. These coordinates 

 are shown in the figures (figs. 1 and 2). 



Fiff. 1. 



Fi£. 2 



r(^>cos9+c»>) 



The method we shall adopt will be to get expressions for 

 the kinetic energy T and potential energy W of the system,. 

 and then apply Lagrange's Equations. 



The shaft is supposed left to itself, i. e. no power is supplied 

 and there are no frictional resistances. 



■§ 3. To get the expression for the kinetic energy T, we 

 shall first consider the motions of the flywheel. The centre 

 of the free end of the shaft has motions (fig. 1) given by 



(a) z in the plane of the paper, 



(b) zco at right angles to the plane of the paper, 



(o being the uniform speed of rotation of the shaft. The 



