518 Dr. C. Chree on the Whirling and 
(c) If in the problem treated under (b) we assume the 
effect of I’ small to start with, we may employ the simpler type 
for BC,y = _n(2clx+ 3ca? —2%), (8) 
» BA, y/=—n(e/l)(2Pa' —3le? + a”). J 
If » were replaced by gM/6HI this would represent the 
bending of the shaft under the weight of M. 
Treating I’ as small, we find by assuming 7 « cos kt 
(k2-+ w”)4Mec2(e+1)?+ (k2—o*)I!(3e + 21)2=12El(e+l). (9) 
When I’ is wholly neglected, this agrees with (6). 
lor the critical angular velocity, we have from (9) 
Ca ROR Oe I’ (86+ 21)? Vo 
= Mea) U7 Matte+08 J) eae 
which agrees with (7) when (I’)’ is neglected. 
§ 21. (d) Loaded shaft of appr eciable mass. 
On Dunkerley’s hypothesis the critical angular velocity is 
given by 
1/@’?= 1/0, + 1/@.’, 
where @, is given by (1) and @, by (5) with & omitted. So 
long as the effect of I’ is small, and ¢// does not exceed 1/4, 
the following approximation is deducible from (3) and (7) 
SUNG tory pe 6) Me(e+l)  U(8e+2t)? 
wo 97° Mts "SE - 12a 
(e) If, while assuming the displacements (8), we allow for 
the mass of the rod, we replace (9) by 
| ‘ Do jaa 
(A? + @’) E AC (e+ Di+ep4 [?(l? + c?) + 55(P +e) 

(11) 
uit me b+ k?—o?) (3c + 21)°=12E1 (c+). (12) 
Putting k=0 we have a form of the critical velocity 
equation in which allowance is made for the inertia of both 
shatt and load. 
(f) As an alternative to (c) we may assume a type of dis- 
placement answering to the form taken by the shaft when 
bending under its own weight, viz. 
for BC, 
y =nila(4?—P) 4+ 6e?x? —Acw? + 2} ; [ (13) 
for BA, ht 
y= nf —3 a! + 6Pa? —Al legs baie (2/1) (? —c?) (2Pa! — 3la? + a?) $ | 
