﻿234 Dr. J. Morrow on the Lateral Vibration of Bars 



When the bar is of uniform sectional area the complete 

 integral of (1) is well known to be 



y = A cosh ax + B sinh ax + C cos fix + D sin fix. . (2) 



In the cases dealt with in this paper the cross-section is 

 assumed to be constant. 



Section II. Unloaded Massive Bar, Supported at each End 

 and subjected to an Axial Pull P. 



§ 3. In the consideration of a special problem we may 

 take, instead of equation (1), a particular differential equation 

 expressing the conditions of that problem only. This method 

 will be adopted as it brings out the physical significance of 

 each term very prominently. The solution for a bar supported 

 at each end has been otherwise obtained. (CJ. Rayleigh's 

 ' Sound/ vol. i. article 189, in which rotatory inertia is taken 

 into account.) Taking the origin at one end, the particular 

 form of the differential equation is 



d}q x C l i x 



~ EI ^ = ""2j W^+l pcoy z {x-z)dz-'Pij 



where y z — deflexion at a distance z from the origin, 

 /= length of bar. 

 If y 1 = displacement at centre of span, the solution is 



. 7TX 



and ;; 1 = Ei7r 4 + p/v 2 



y 1 ~ pul* ' 



from which the frequency is obtained. 



When the force P is compressive vibration is impossible if 



^ =0, that is when 2 



Section III. Clamped- Clamped Unloaded Massive Bar. 

 Axial Bull = P. 



§ 4. When the ends are clamped, using equation (2), the 

 conditions at the ends and centre give 



x = 0, 





y = 0, 



.: C=-A, 



x = 0, 





ax 



•• D =-? B > 



X = /, 



y= 



dx '' 



and x=rz, ~r = ® 

 2 ax 



