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



Section IV. Clampecl- Supported Bar under Axial Pull. 



§ 11. When the bar is clamped at one end and merely 

 supported at the other, the solution can be deduced from that 

 of the previous section. Thus in the final equation of § 7 we 

 may give to i, in succession, the values of the positive 

 integers and at the same time write 21 for I. 



For the fundamental 





FT P P 2 



r ^+ 12-29 ~i---0912- " 



pew/ 4 pool 2 pwEI 



/ is the length of the bar, and the result is still confined to 



PP 

 .small values of ^ T . 



Vj l 



Section V. Bar of Negligible Mass, Supported at each end. 

 Compressive Axial Force = P, having a single load con- 

 centrated at some point in its length. 



§ 12. When dealing with bars of negligible mass it is 

 convenient to assume the positive P to be compressive. This 

 change of notation is therefore made in the remainder of the 

 paper. 



Let the mass m divide the bar into segments a and b. 



The equations are : — 



For.i*<a T7,x^ 2 7 -r. ■• b 



— Jiil -r~ = r a — mil T x, 



dx 2 J Ja l ' 



for x > a wT&y -p " a n . 



dJ y ~ imja 2 ^ '' 

 ■ 

 These have respectively the solutions 



ii = — x + B sin nx — A cos nx, 



u = -^ (x — I) + D sin nx — C cos nx. 



.J n & \ 



Where a P my a b , , my a a 



u = wr c= ei p andA= -Eir 



The conditions at each end are 



x=0, y=0; /. A = 



jj = l, ,y = 0; .'. Dsmnl 



=Ccosnlj 



(13) 



