Vibrations of a Rod of Varying Cross-Section. 87 



and on writing u = t/rj n+1 J this becomes 



g^|±<-^<-» m 



which is the Bessel's equation of order n+1. 



If we take one end of the bar to be at z = 0, and consider 

 that end to be free, then, since the displacement there must 

 be finite, we only need the Bessel's functions of the first 

 kind in the solution of (9), Hence 



t = AI H+1 {fi) + BJn+i(y) 9 .... (10) 



or u = A<f> n (x) + B^r„(#), (11) 



where 



<M>)=-4iW2^), M<e)=4 E Jn + i(2Sx). (12) 



X 2 a? 2 



3. Clamped-Free Bar. 



Suppose the rod to be clamped at z — l. The conditions to 

 be fulfilled at a clamped end are £ = and 9f/^^=0, which 

 in our case reduce to u=0 and du/dx=0. From (12) we 

 then have 



A0„O& 1 )+B^ H (tfi) = O, .... (13) 



A ^M +B »W =0 b b ^ (1 

 ax ax K ' 



where ^ is the value of x corresponding to z=L Now 

 according to our notation <t> n -u tyn-i are certain integrals of 

 the equations 



d 2 <j) n -i . , . -.x d$n-i , 



(15) 



dx 2 dx 



*° = TT and ** = — 3*~- • • • (16) 



Hence (13) and (14) give 





S-. • • • (17) 



