Transverse Vibrations of Rotating Shafts. d41 
This is analogous of course to Dunkerley’s hypothesis, but it 
is far from amounting to a proof. Hven if we assumed that 
what is true of transverse vibration frequencies is true of 
whirling velocities, we should have to prove that the addition 
of pulleys at different parts of a shaft is equivalent to varying 
the load without affecting the forces of restitution. 
§ 45. The following investigation would seem to show 
that the result is not in general strictly true, though it may 
be, and not improbably often is, a close approximation to 
the truth. 
Suppose that a massless shaft of length /, supported at its 
ends A and B, carries a mass M, at C(AC =a), and a second 
mass M, at D (CD=c, BD=8), the effect of the moment of 
inertia being negligible in either case. 
Measuring wz from A, and 2’ from B, we may assume the 
following types of displacement—derived by considering 
the bending of the shaft under the weight of the two loads :— 
from A to C, y=n[ M,(6+c)e{P—(b64+¢)?—27} | 
+ Moba(? —b? — x”) 1, 
Cto D, y=nl[Mae'(? -—a-2” 
, y=n|Myao'( anh en ean es) 
+ M,ba(? —b?— x”)], 
» Dito B, y=n| Maz’ (?P—a?—2") | 
+ M,(at+e)a{P —(at+e6)?—2}},. 
Taking as usual for the angular velocity, and applying 
Lagrange’s equations, we find after algebraic manipulation 
1/( +?) =M,{a?(b + ¢)?/3HI/} + Mo{6?(a+<¢)?/3EU}—R, (26) 
where 
R= M,M,(M, + M,)a?6?e?(4ab + 4ac + 4bc + 3c”) 
+ 12EIU{ M,?a7(6 +c)? + M.76?(a +c)? 
+ M,M,ab(2ab + 2ac+2be+e)}. . (27) 
For the critical angular velocity answering to whirling we 
put £=0, and find 
1/w?=1/o,+1/o.°—R, . ovat 120) 
where 
o, =3KU+Myo7(b+¢)?, ow” =3KI/+M,6?(a+e). 
Referring to (9) or (10) §13, we see that w, and , are 
the critical velocities for the shaft when loaded with the mass 
M, and when loaded with the mass M,. 
In order that (28) should agree with Dunkerley’s hypo- 
thesis R should vanish. It is, however, obvious that R is 
positive for all values of M,/M,, and for all values of a, b, 
