The Lateral Vibration of Loaded Shafts. 



305 



pointed out, however, that an adequate mathematical treat- 

 ment of the general case of a loaded shaft is a matter of 

 considerable difficulty, and that the discussion which follows 

 does not pretend to bo more than illustrative. 



2. For the sake of simplicity in illustrating the character 

 of these vibrations we will consider the case of a light 

 uniform shaft supported freely in bearings at its ends and 

 carrying a mass m at the centre of its span, the mass centre, 

 however, being slightly eccentric by a distance a from the 

 elastic centre of the shaft. 



We will suppose the load to be of the nature of a thin 

 pulley or disk of negligible moment of inertia. Accordingly 

 we are dealing with simple lateral vibration, and are not 

 concerned with oscillatory vibration about a diameter. 



The conditions given permit a statement of the problem 

 in a simple form. The motion of the cross-section at the 

 centre of the span in its own plane need alone be considered, 

 subject to (a) a restoring force varying as the distance of a 

 point known as the elastic centre from the axis of the 

 bearings, (b) a damping force, and (c) the disturbing effect 

 produced by an impressed rotation in its own plane combined 

 with the fact that the centre of mass is placed eccentrically 

 with respect to the elastic centre. Thus the problem may be 

 viewed as the motion in the plane XY of a disk on which is 

 impressed a constant angular velocity g>. 



In the cross-section in fig. 1, let be the intersection of 

 the axis of bearings with the plane XY. Let x and y be the 

 co-ordinates of E the elastic centre (which is also supposed to 

 be the centre of figure) of the shaft at any instant ; and let 

 M be the position of the centre of mass at the same instant. 



It is given that the shaft rotates with angular velocity &>. 

 This angular velocity determines the angle (cot) which any 



