Transverse Vibrations of Rotating Shafts. so 
This is not quite correct; the true limiting value is 7, exactly 
as in (a) Case 2. 
Treating ¢// as small though not negligible, I find for the 
next approximation 
pla Lee ote he's 5 (2) 
Whence, substituting its numerical value for 7, we have 
2 oD A 
for the critical angular velocity 
w?=97-41(EL/opl')(1—6°6e/P). . .. (3) 
This is satisfactory so long as (¢/l)*(7°/6) cotha is small 
compared with unity. 
A way of treating this problem by an assumed type of 
vibration will be found below in (f/f). 
§ 20 (b) Mass (M, I’) on overhanging part of massless 
shaft at distance c from the nearest support. 
Formule of the Rayleigh type are 
oy = 3st OO OE + 30) — 22(-6 80) (:) 7 

c(dc+A4l) de+4l 
4 
cO—3z (4) 
ipa : Ape f 12 13 
” BA, y Hoe 21? x — 3la -+-2@ je 
By Lagrange’s equations we find for the frequency equation 
( det l a c+ \ 
{ Me a) 19 al aay f { V(2—0%) —19BI SF" ioe 
_ 36(E1)2(3c + 21)2 h 
> Lee a 

~~ 
Putting £=0 we obtain an equation for the critical angular 
velocity w which is identical with Dunkerley’s (J. ¢. p.. 314). 
If in (5) we neglect I’ altogether, we find 
24-0? =3HI=Me(e+). 2...) 2 6) 
Proceeding to a second approximation, retaining only the 
lowest ary “of I’, we find for the critical angular velocity, 
putting = 
ted 3KI (3¢ + 21)? 
ses F as ii He+l)? rf eG 
This last result will not be satisfactory when e/l is very 
small, or the load very close to B; but under these con- 
ditions 1/w? is very small, so that on ‘Dunkerley’ s hypothesis 
the load has but little aS) influence on the critical velocity. 

