932 Dr. C. Chree on the Whirling and 
The least root of (2) (see Rayleigh’s Sound, art. 174) is 
pl=4°73004, 9 
which gives for the critical velocity 
a =500°6(HE/cpl). 2 2 
(6) In the above case of no carried load, an approximate 
frequency equation is deducible from the simple type 
y=no(l—a). 
Replacing n by gop/24EI we should have the bending of 
the shaft under its own weight. 
From the kinetic tr cgten ert we easily find 
k?+o@7?=504(ELopl'), . . 2 ae 
eiving for the critical velocity 
oe =904(El apt’)... 2 
The values given by (4) and (7) for » differ by only about 
0°3 per cent. 
§ 34. Load (M, I’) on massless shaft. 
(c) If the load be at C (AC=a, BC=b) we have for 
Rayleigh type displacements, measuring # from A and gz’ 
from 
in AC, y=(82—a8) (w/a)? + (a9 — 22) (w/a), 
in BC, y= (32 + 60) (a/b)? — (8 + 22) (a'/b)? 
Applying Lagrange’s equations we have 
{M(k? + w”)— 12K (a-? +-b-*) § {1 ? —@”) — 4 EI (a-! +874) 
=36(HI)7(67 —a°*)? 9...) 
Omitting £’, we obtain for the critical |velocity a result 
agreeing with Dunkerley’ s (lc. p. 887). 
(9) splits into factors, representing pure transverse and 
oscillational vibrations, lien b=a. In this case we have 
for the respective frequencies (cf. (11) of §26, and (13) and 
(14) of §42) 
(3) 
k? = —w’ + 192K1/MP 
sei el 
R= of +16HI/Tl. | 
If we omit I’ altogether in (9) we obtain for the critical 
velocity | 
o’=3hIl/Mad*. 
As a second approximation, when I’ is small, we have 
w= (BEI/?/Ma*b?) {1 + 91'(a—b)?/4Ma?b?t. . (12) 

