746 On the Transverse Vibrations of Prismatic Bars. 



In order to estimate the influence of the shear upon the 

 frequency of the vibrations, let us consider the case of a 

 prismatic bar with supported ends. The type of the 

 vibrations may be assumed to be given by 



. mir.v 

 y=Y sm-^cos p m t, (9) 



where I represents the length of the bar, and p m is the re- 

 quired frequency. By substitution from (9) in equation (8) r 

 we obtain the following equation for the frequency : 



P 



mVk' 2 / E\ k 2 p 4 



,- p-[ 1 +xo)^ + fi^' , ' =0 - (10) 



If only the first two terms on the left side of this equation 

 are retained (this will correspond to the equation (1)), we 

 have 



9 9 9 



P TO = a -p- = X r ' ^ ^ 



where L= — represents the length of a wave. 

 m x 



By retaining the first three terms of equation (10) (i. e. by 

 neolecting the terms which involve X), we find 



v ="^l_ l7 ^ (19) 



approximately : this result corresponds to equation (2), 

 where the rotatory inertia is taken into consideration. 



By using the complete equation (10), and neglecting small 

 quantities of the second order, we find 



approximately. ^ 

 Assuming the values 



X=f, E=|C, 



E A 



we have rn — 4, 



and hence we see that the correction for shear is four 

 times greater than the correction for rotatory inertia. The 

 value of the correction of course increases with a decrease in 

 the wave-length L, i. e., with an increase in m. 



Yougoslayia, Videui. 

 Summer 1920. 



