510 Dr. C. Chree on the Whirling and 
If for 7 we substitute goo/24HI1, we have the displacement 
which the shaft would experience if bending under its own 
weight. or the present purpose we assume 
n & cos kt, 
where 4/27 is the frequency of the lateral vibrations. 
Neglecting the inertia of the motion of cross-sections 
relative to their centres of gravity, we find from (5) 
T=3(7 + 0%?) ‘opt, | 
= . (6) 
whence, by Lagrange’s equation, 
k? + @t= 12°46 (Elicpl). . 2 a 
This gives the frequency of vibration for any assigned value 
of wo. 
The critical angular velocity answering to whirling is 
that for which / vanishes, or the motion becomes unstable ; 
it is thus given by ; 
wo’ =12:46(El/opl*). . .. 7 a 
The values given by (3) and (7) for the critical value of o 
differ by less than $ per cent. 
§ 8. (c) Mass M, inertia I’, at end of massless shaft. 
Assume (c/. Rayleigh’s ‘Sound,’ art. 183) 
y = (82—16) (2/1)? + (10—22) (x/D3, . . . (9) 
where z and @ are the values of y and dy/da (which may be 
regarded as the inclination of the shaft to its undisturbed 
direction) at the point of attachment of the load. 
By Lagrange’s equations, or otherwise, we obtain for the 
frequency 7 
4 (k? + @2)M—12EI/? hf (2 — @?) /—4 BI /lt =36(EI/?)?. (10) 
For any assigned value of w, (10) gives two values of /2, 
answering to two different types of vibration. Only one of 
these—--which answers normally to the smaller value of k2—is 
properly speaking of the lateral type. 
For the critical angular velocity answering to whirling, 
we put £=0 and obtain a quadratic equation for w?, identical 
with that obtained otherwise by Dunkerley (J. c. p. 304). 
One of these values of w? is negative, and has no applica- 
tion to the present problem. | 
It, as in Dunkerley’s experiments, I’ has but little effect, 
a first approximation to the desired value of w?—obtained by 
omitting lL’ altogether—is 
@=3RYME SS. 
