Vibrations of a Hod of Varying Cross-Section. 



Third Mode (# 1 = 57*380). 

 M=C{664-841^i(a?)+ -000414102^ (a?)}. 



101 





X. 



zjl. 





6-59385 

 17-70786 

 33-93229 

 51-29925 



•114916 

 •308608 

 •591361 

 •894026 



Loops 



1017632 

 23-83643 

 41-75737 



•177349 

 •415413 

 •727735 





9. Free-supported Bar. 



Finally we may briefly notice the case in which the bar is 

 supported at z = l. As before we have 



M=A<£„(.??)+Biki0c). 



The conditions to be fulfilled at a supported end are £ =0 

 and coK 2 'd 2 SI^2 2 = 0, which in our case reduce to 



u = and ^ +3 S=0, 



and therefore yield A£ n (#)+B^r„(a?)=0, .... (38) 



- A< S +B 2" =° w 



Eliminating A and B from (38) and (39) we have 



d*yfr n d 2 (f) n 



dx 2 



da 



which if n is integral gives as the equation from which a? lt 

 and therefore X, is to be determined, 



= 



x l 



+ 



X q 



(n + 2)!(rc + 4)! l!(n + 3)!(n + 6)! 2 !(n + 4) ! (w + 8) ! 



. . . (40) 



10. To obtain the case of the wedge-shaped bar put ?i = 0. 

 Equation (40) then becomes 



= 



X' 



T- + 



%* 



2!4! 1 ! 3 ! 6 ! 2 ! 4 ! 8 ! 

 the lower roots of which are 



10-902, 24-631, 43-204. 



