Vibrations of a Rod of Varying Cross- Section. 

 To find the points of inflexion we have 



9a 



• 



2^x. 



X. 



d-<j> 



dx 2 ' 



d 2 rp 

 dx 2 ' 



dnc j 



J6-38 

 1 6-40 



T9-62 

 \9-64 



101761 



10-24 



23-1361 

 232324 



1-38222 

 1-389101 



10-8118 

 10-9617 



+ -00000147866 

 -•000180292 



-•000317421 

 -•000270646 



-•257026 



+1-01470 



+ •29345 

 -•06087 



The required points are then given by 

 0=10-18901, 23-21586, 

 z/l= -339407, -773348. 

 The form of the rod is shown in fig. 3. 



Fis-. 3. 



In the fourth mode x x = 49*763, 



W = C{-155668 $£x) -143695 ^<>W} 



The positions of the nodes, loops, and points of inflexion are 

 then as follows: 





X. 



zfl. 



Nodes 



3-67055 

 12-30668 

 25-80624 



•073762 

 •247306 



•518582 





Loops 



6-59392 

 17'70437 

 3407126 



•132506 

 •355774 

 •684669 







10-17609 

 23-84908 

 41-37971 



•204491 

 •479254 

 •831536 



5. By putting n = l we obtain the case of a very sharply 

 pointed pyramid on a square base. It can be readily shown 

 that the same analysis applies to the case of a circular cone. 



