Uniform and Varying Sectional Area. 115 



If y z be the displacement at a distance z from the point 

 72 

 chosen as origin, so that -~ is the acceleration, and if y 1 be 



the displacement of any chosen point (say that at which the 

 displacement is a maximum), then the above statement may 

 be expressed by the equation 



d % L - dh ^i y 

 dt* \ Jz '~ dp yi 



y=^V* (1) 



Jz V\ 



This is equivalent to the usual relation in frequency problems 

 denoted by 



y = coskt.f(x). 



Now, if we consider the reversed effective forces due to the 

 acceleration of an increment of the length of the bar, we 

 have, by the ordinary Euler-Bernoulli theory, that the couple 



di/ 

 at any section in the direction tending to increase -j- is equal 



to the product of the curvature and the flexural rigidity at 

 that section. That is, in view of the approximate straightness 

 of the rod, 



da* EI' [Z) 



where M is the couple due to the reversed effective forces, 



E= Young's Modulus, 

 and I = Moment of Inertia of the cross-section about the 

 neutral axis. 



Since M is a function of the mass per unit length, the end 

 forces, and the accelerations, we can express it in terms of 



J— y z and the density and cross-sectional area of the bar. And 



#1 



assuming for y z a hypothetical type of vibration satisfying 



the end conditions, we can insert the value of M thus obtained 



in the equation (2). Solving for y we have a second 



approximation to the centre-line of the bar, and can calculate 



the period by 



N =i\AJ • ••••'•• 0) 



The next approximation consists in using this solution for y 



12 



