124 Mr. Morrow on Lateral Vibration of Bars of 



8. Clamped-Free Bar of Varying Depth. (d=A,v.) 



The exact form of the equation first assumed for the type 



of vibration is immaterial to the final result, but, by choosing 



a suitable equation, the labour involved in obtaining this 



result to any required degree of accuracy is reduced. 



d 2 v 

 In the present case (as, indeed, in the last two) ^% is not 



necessarily zero when «r=0, for there the area of the section 

 of the bar also vanishes. 



Hence we may start with an equation 



which, inserted in 



gives 



»=»{ 1 -T + i)> • • • • ( 10 ) 



" S = E& fS 6A *(*-%. • dz 



_ c Em __EiL(<>_ 2x ,.«A. 



o 4 



■• -y 



and _ £ _ 60 EA 2 



In the second approximation 



+ -001,190,48^), 



.-. -y=^^|L^-1742064 2 4 --4468254/ 3 .i- + -4027ZV i 



- -16 lx' + -0416 x i - -005y + -0003968 



P)' 



_?--^, EAi 



2-312; 



and in the third approximation 



~% = ^W^ P H' 290U /4 -' 37235 z x + ' 2014 /2 * 2 



-•05 lx 3 + '0099ff 1 - -0001 j\ 



