514 Dr. C. Chree on the Whirling and 
but do not neglect the mass of the shaft, we replace (8) by 
{ (I? + 0) (M+ spo! )— SEL (a? 40-8 ; x 
9 2 ! Dy) 2 3 
: { (i081 + (kh? + 7) es *4+6°)—dHI (a7! +6- » t 
= (01)? } 3Bla*- 24 2 (B+! Jeo >) Se 
This may be expected to give best results when the load is 
at or near the centre of the span, as the assumed type of 
displacement, which answers to the load only, is then nearest 
to that natural to a massive but unloaded shaft. 
If the load is at the exact centre (15) gives 
ie +o? = 48H { Mn+ qoet be ha 

When M is neglected this gives 
+o? =93' 8H /epl*... . 2) eee q 
Putting k=0 in (16) and (17) we have of course the cor- ; 
ee critical angular velocities. The value obtained : 
from (17) is.a ae io approximation for the case of an , 
unloaded shaft (cf. ( ’ 
§ 15. (g) When Vr ‘ ‘small, but the load M is not near the 
centre of the span, and is of the same order as the mass m ; 
of the shaft, better results are obtained from the following 
displacement types :— 
for AC, y =na{m(l—a)(P? +lx— x’) Meee 18 
for BC, y'=na' {mila (UP We a) eae ey see 
If 7 were replaced by g/24HI/ this would give the bending 
of the shaft under its own weight and that of the load com- 
bined. Assuming 7« cos At we obtain the frequency equation 

(FP +o@°)A+ (P—o’)/B=C, . . . (19) 
giving for the critical angular velocity 
w'=C—(A—VB); (0 4). ae 
where tor brevity 
a ee ey ){ 452% +76(S) +36(") ; ] 
31m iz | 
96 (M2? (ab ab ab 40320 /M ae 
+31Gn) Ce) {8esatp +107 (jr) yt (mn) (= )> | z 
P 630 (a—b)* 1 gab Ayok 8M ae ‘ re{ 
ee ale ( P P 
(- 3024 EI Mab ab M \? sab | 
ag oa pit eer a (i+ —&P )+40(7) (i) \, ) 
