1890.] 

 Also 



vibrations of a revolving cylinder or bell. 



105 



W 2 = \ix<r {(a + w) 2 - a 2 } ad6 



i 



sw r 



= I \fxaa (2aw + vf) dO; 

 io 



= I [i<ra 2 8wdd + fiaaiv&vdO (5). 



Jo Jo 



Finally for the energy of bending we have if A (1/-R) is the 

 change of curvature, 



where, for a cylindrical shell of thickness 2/i 



= (E+T)I. 



SF =?/ 2 '(^ + 1 



and for a ring* 

 We readily find 



The variational equation of motion 



becomes, therefore, on slightly rearranging the terms, 

 r2w 

 0= [aa 2 (- eo 2 + fi) + T] SwdO 



Jo 



+ I \aa(v + 2(ow) Sv + aa (ib — 2coV — w 2 w) Sw 



.(6). 



Tdw . 

 a du 



T/dv d 2 w\ 



I ^ — r-^ 1 + fiamu + -g 



£ (d 



+ 1 )w 



8w\ dd. 



a\d6 dd 2 ) ' r ™ ' a 3 \d6 2 



For the undisturbed motion we have v = 0, w = 0, and the first 

 line of the above expression gives 



T=<ra 2 (a> 2 -tM), 

 as already found (1). 



For the oscillations, we must assume with Lord Eayleigh that 

 the extension vanishes to the first order, so that 



dv \ 

 d£ 



w = 



and 



o dSv 



.(7). 



* See Lord Eayleigh, Theory of Sound, i. p. 242. Here E denotes Young's 

 modulus and //.' Poisson's ratio. 



