524 _ Dr. C. Chree on the Whirling and 
x from A to C, and 2! from B to C, we have for the Rayleigh 
type of displacement 
for AC, y=(3¢—a8) (v?/a”) + (a8 —22)(x?/a°), 
for BC, y!=4(382 + 66) (a/b) —3(08 +z) (2? /6°) Agee: 
By Lagrange’s equations we find 
{MI (en yooh ee t {rq o) Ea | 
— 2b? 
= (3EI)? “(es3 aa) i ao 
When £& is omitted this agrees with Dunkerley’s equation 
(l.c. p. 321). Hquation (10) splits into two factors when 
ajb= f2=1-414. 
In this position of the load the frequencies of what may 
be called the transverse and the oscillational vibrations are 
respectively given by 

W2= —o? + (51436 Mlene amends (11) 
R= w+ (745 V2) (EIT) = o?4+14:14(BI/ID. 
If we wholly neglect I’ in (10), we have for the critical 
velocity 
wo” = 12H13 =~ {Ma*b?(3at+4b)}3; . . (12) 
while retaining I’, but neglecting (1’)?, we have 
er wee ee 91’ (a? — 267)? 
Oi Ma’l?(3a + 4b) {i+ Ma?b?(3a + 4b)? Te 
(d) If we assume I’ small to commence with, still neglecting 
the mass of the shaft, we may take 
in AC, y=n{22(3ae?— 2°) —a?(2a + 3b) (3la?—2°) f, | 
in BC, y=n{307bPx' —a?(2a+3b)2"}. J 
When 7 is replaced by Mg/12KI/? we have the bending of 
the shaft under the weight of M.- 
For the kinetic problem, we find for the frequency equation 
(k? + w*) Ma*b?(3a + 4)? + (K?—@?) 9 a(a?—20")? 
=12HI0(3a+4b), . (15) 
and for the critical angular velocity 
oa Sige ee SOI = 207)? ae 
~ Ma*b?(3a + 4) Ma?l?(8a+4b)” 

(13) 
. (14) 


(16) 

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: 
a) 
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a gue igewed 4y0* 
1 EP ee oe) ee Ty < 
a a 

