310 Prof. H. H. Jeffcott on Lateral Vibration of Loaded 



in the opposite direction by revolving counter clockwise at 

 the same speed. Thus the true heavy or light spot lies 

 midway between the centres of the markings in these two 

 cases. 



In most cases, when balancing, the machine is operated 

 by running above the critical speed, thereby marking on the 

 light side. 



It is to be noted, however, that the cases of balancing met 

 with in practice are usually complicated by having several 

 masses along the shaft instead of only one as considered 

 above, so that the resultant displacement at any section then 

 depends on the joint action of all the loads, according to 

 their several effects at that section. 



(>. It should be pointed out that, at speeds appreciably 

 below the critical value, the phase-change due to damping 

 is small. The introduction of the coefficient b in the equation 

 of motion is mainly of interest in showing how the mass 

 centre changes from being outside to being inside the 

 elastic centre, relatively to the axis of rotation, as the speed 

 rises from a little below to a little above the whirling speed ; 

 and also that the amplitude of vibration at the critical speed 

 in the ideal case under consideration, in which no restraint 

 to excessive vibration is assumed to be imposed by the 

 bearings, would not become indefinitely great. 



Returning now to the amplitude of the vibration, which is 



V(c-m~oo 2 ) 2 + b 2 <o 2 

 its value is not seriously altered by the omission^f b except 

 in the close neighbourhood of the value w= \f — . 



At this critical value of g>, the effect of b is to diminish 

 the amplitude and prevent it increasing to an indefinitely 

 great value, as has just been stated. 



7. If 6 = 0, and also m<o 2 = c, the equation of motion is 



x + (o 2 x = aw 2 cos tot, 



of which the solution is 



x — A sin (cot + ol) + \awt sin cot , 



where A and a are arbitrary constants. 



The amplitude of the forced vibration would in this case 

 increase continually with t. But the damping b is not zero, 

 and the amplitude therefore does not increase indefinitely. 



