for determining Frequencies of Lateral Vibration. 421 



suggest that — given accurate draughtsmanship, and assuming 

 the validity of the theory of thin rods — errors will not ex- 

 ceed 1 or 2 per cent, in any practical problem. 



Stated mathematically, our object is to determine the 

 forms of certain curves of deflexion (the " normal modes " 

 of vibration) which are associated with any given shape 

 of rod. These curves form a family which is defined 

 by a certain differential equation, and they are subject to 

 certain "conditions of constraint" (expressing the effects of 

 clamps, journals, etc., in the system under consideration), 

 which have the common feature that they leave the 

 magnitude of the deflexion unrestricted, whilst defining 

 the manner of its variation along the length of the rod. 

 In the problem of " whirling," each curve of deflexion has 

 the property that it can just be maintained, when the shaft 

 is rotating at some definite and appropriate speed, by the 

 centrifugal forces acting against the elastic restoring forces : 

 in the vibrating bar, the instantaneous deflexion at every 

 section of the rod is compounded of one or more "normal 

 modes of vibration ," any one of which may be regarded as a 

 curve of deflexion which varies, in absolute magnitude, as 

 some appropriate simple harmonic function of time. 



The differential equation which governs the curve of 

 deflexion may be obtained as follows : — 

 Let 



p be the line- density of the rod at any section, in 



pounds per foot run, 

 2?( = EI) be the flexural rigidity at the same section, 



in pound- (foot) 2 units, 

 and x be the distance of the section from one end of the 



rod, measured in feet 

 (so that p and $3 vary in some specified manner 

 with x) : 

 and let?/ be the instantaneous deflexion of the shaft, in feet, 



at the section x, 

 and n be the number of vibrations — or of revolutions, in 



the problem of the whirling shaft — per second 



(so that our problem is to determine the value of 



n and the manner in which y varies with x) : 



then the elastic restoring couple M, which acts at any section 

 of the deflected shaft, is given by the usual equation 



M+»|^«=0, (1) 



OX" 

 Phil Mag. S. 6. Vol. 41. No. 243. March 1921. 2 V 



