﻿238 Di\ J. Morrow on the Lateral Vibration of Bars 



necessary conditions are not fulfilled and this solution is 

 inapplicable. 



§ 9. A solution of the present problem free from the 

 restrictions of the previous paragraphs may be obtained by 

 the method of continuous approximation *. 



Thus in the equation 



-El3.p.J, {j^-^-^'f «,,} -P,. M 



■ — ,-%>(*-¥♦*) an 



Integrating twice, 



-EL/= ^p-16 x 10- 3 A-785715 JV-2-7 lV + 2'7 a 6 



- 2-380952 ~ 4- '595238 |f) 



~-f A (--01^V + '083^--l|+-03^ 

 and _ jji _ 494'74(EI+ 025P1 2 ) 



from which the approximate frequency may be determined. 

 Proceeding to a second approximation we obtain a simple 

 but lengthy expression for the vibration type, viz. : — 



-EIy = ' 00 ™ 158 (l + i ^)[p«y 1 (3-5573 JW-5'5113 IV 



+ 4-9603 /% 6 -3'3069/V + -5511 ^ 10 --3006^- +-0501^) 

 + Py/42-99 IV - 1)6-1377 IV + 46-2963 IV - 49-6032 x 8 

 + 33*0688 y -6:6138^°)] + ^10~ ^pcoy J -^06S5nV 

 + 1-488093 IW - 1-38 Pz 5 + -496032 a? 8 - -330688 ~ + '066138 y) 

 -Py/3-968250^r 2 -13-8^ 4 + 27-7^ 6 --.23-809524y 



and ' + 5-952381 Jj], 



_y, __ 500-4(EI) 3 +12-1526EIP/ 2 --01266P 2 ^ 

 y x *~ (EI--0003838P/ 2 ) / ooj/ 4 "" ' 



* See Phil. Mag. ser. 6, vol. x. no. 55, p. 113. 



