a 
Transverse Vibrations of Rotating Shafts. 527 
critical velocity 
ko = (4nso re) ibieam oo a (11) 
w= 25050 (lil iopir pent. Coe (12) 
(c) As an alternative type to (7) let us take 
for AO, y= —n{x'—4a°x—C (a? —3a°x) — De}, 
for BO, y/= nfe— 40x! —C! (#3 —3Ba’) — ae (13) 
where now 
C= (3a? + 4a7b +b?) =2a(a+b), C= (a+ 4ab? +36?) +26(a+d), 
D=ab(a?+0’)/(a+d). 
The above displacements answer to an imaginary gravi- 
tational force oppositely directed on the two sides of O, 
By Lagrange’s equations we deduce 
fo 
ats Ph (G59) 4 ae(Ca" 4 wens 
+3 1 D(a 48%) 5, Ca’ +0) +! ‘D(a’ +h") D(Ca' +0) |. aay 7 
Pee. e,. we have the critical angular velocity. 
When 6/a is small (14) agrees with (8) in giving (9) and 
(10). Kor b=a, however, it gives the widely different 
result 
k? + w= (48 x 63/31) (HI /opa*), . . . (15) 
whence for the critical velocity 
or 97-ob Ci Vapar)s.«+. ‘1 2 Leal 
This is identical with (7) of case (2), which applies to a 
single span of length a, and is in close agreement—as it 
should be—w ith ( 3), when b is replaced by a. The diver- 
gence of (12) te means (cf. § 24 and Table IV.) that its 
assumed type of vibration answers not to the fundamental 
note but to an harmonic. 
§ 29. Mass (M, I’) on massless shaft. 
(d) Supposing the load at C, between O and A, ata dis- 
tance c from O, Rayleigh type displacements are : 

; _ 26(32— 6) “) Oc(26 + 3¢) —2z ee i a 
re ee ey ay aa 4b+3c ~~ a 
‘ nog t (a—c)@ sult —c)0+ & 2 | , 
for AC, y! = Fiasscay i x sane ies 7 (1 : 
. 7 2(O—32) (Ba! La | 
= gap DP oy ites (~ ae ee 7) J 
where z and «” are measured from O, and 2’ from A. 
