Transverse Vibrations of Rotating Shafts. 523 
CasE 4 (Dunkerley’s Cases IV. and XII). 
§ 26. Shaft fixed in direction at one end (A, ea) and 
supported at the other (B, 2=/). 
A B 
oe mers 
(a) Unloaded shaft: Euler- Bernoulli solution. 
The displacement is given by 
y=a{ (cosh wx—cos wx) (sinh pl + sin wl). 
—(sinh we—sin wu) (cosh wi+ecosulj)}, . (1) 
where @ is a constant. 
The equation determining w (cf. Dunkerley, l.c. p 294) is 
Goth pl—Gob ple. gy 
The least root of this (Rayleigh’s ‘Sound,’ arts. 180 and 
174) is 
pl or9200... age Oe eT 
Answering to which we have tor the critical angular velocity 
ae — soir Cluliepl |... 2 queen (4) 
(6) Instead of the Huler-Bernoulii method, we may 
assume for the unloaded shaft 
y=nx (l—2) (31-22). og eee) 
If » were replaced by gop/48HI, this would give the 
bending of the shaft under its own weight. 
For the dynamical problem we find 
T= sap (1° + on”) (19/630)2, 
V=49' (36/5) EIB ; a me 
whence we have for the frequency equation 
i? + a = (4536/19) (EV/op ty 3 2 ea 
and for the critical angular velocity 
ers 20 1 Maat eh. ce at ot eae 
The value of given by (8) is only 0°2 per cent. in excess 
of that given by “the exact equation (4). 
(c) Mass (M, I’) on massless shaft. 
Supposing the mass at C (AC=a, BO=4), and measuring 
