Transverse Vibrations of Rotating Shafts. 533 
(d) If we can assume I’ small to begin with, we may 
replace (8) by 
in AC, y=n{ab?(a+ b)a?—1b?(3a+ b) 2°}, 
in BC, y/=n{a*b(a + b)x2’?—4a? (a+ 3b) 2} 
Applying Lagrange’s equations we find 
(A? + w) $Ma*d? + (h?—w*)ab(a—b)? = 4 El (a +5)’. (14) 
For the critical angular velocity, writing 1 for a+b, we 
have 
b,c) 
w° = (dHI?/Ma*b*){1—91'(a—b)?/4Ma??}-! . (15) 
Neglecting I’ this agrees with (11), neglecting (1')? with 
(12). 
§ 35. (c) Load (M, I’) on massive shaft. 
7a Dunkerley’s hypothesis we have 
1/@ = Ver? +1/o” 3 
where @; is given by (4) and @, by (9) with & omitted. 
When I’ is small we thus obtain 
EN epe Mab? — 3: V’ab(a—b)? 
a 500-6KI *3mIe a mie - ~ ‘4 
I have not worked out a frequency equation based on an 
assumed type of displacement, but it would present no 
difficulty. The displacements would combine terms of the 
types (5) and (13) according to the relative masses of the 
shaft and load. 

GENERAL CONCLUSIONS. 
§ 36. In every case here treated when the effect of the 
moment of inertia of the load has been small—as was true 
invariably in Dunkerley’s experiments, and probably often is 
in practice—the frequency equation has proved to be of the 
type 
== ES 7 en eer em Ti 
where K/27 is the frequency of the fundamental transverse 
vibration of the system when not rotating. There would 
thus seem grounds for supposing that a for rmula of type (1) 
will often prove a close approximation to the truth. When 
this is the case, we can arrive at a close approximation to the 
velocity answering to whirling without endangering the 
shaft by actually pushing the velocity to this point. All that 
is necessary is to determine the frequency of the lowest 
natural transverse vibration in the shaft when not rotating, 
and when rotating with any convenient, velocity @. 
If these frequencies be K/2a and &,/2a respectively, then 
