508 Dr. C. Chree on the Whirling and 
§ 4. As stated above, whirling is not really a case of:co- 
incidence of period between a vibrating system and disturbing 
forces. A rotating shaft may, however ;—like any other 
shaft—be acted on “by periodic forces which tend directly to 
set up lateral vibrations. In considering the effect of any 
such forces, it must be borne in view that what one has to look 
to is the frequency of the lateral vibrations of the shaft as 
reduced by the rotation. The possibility of forced vibrations 
of this kind is an additional reason for considering the effects 
of rotation on the period. 
$5. In the main I shall follow Dunkerley’s classification 
of the principal cases of whirling shafts, but shall not number 
separately cases where the shaft is with, and without, a load. 
The cases are determined by the number and nature of the 
supports. 
If x be taken parallel, and y per ae RR Ye to the andi 
turbed position of the axis of the shaft, the bending being 
supposed to occur in the plane xy, clearly at any support a 
70. 
If the end of a shaft simply rests on a support, then on the 
Huler-Bernoulli theory, as the terminal section must be free 
from a couple, 
dyjdeA=0 
At such an end the shaft is said to be “supported.” If, 
on the other hand, the shaft be constrained to retain a fixed 
direction at an end, the second terminal condition is | 
dy dz; 
Ifa shaft is “supported ” at any intermediate point, chen 
clearly y must vanish there, while dy/dw and d*y/da? must 
be continuous. A sudden change of dy/dvx would imply. 
fracture, while a sudden change of d?y/d«? would imply the 
action of a couple at the supported section. ah 
When an end is quite free, resting on no support, both 
stress and couple vanish, and so 
Cylde=Cylae= 
Notation used. 
Ke = Young’ s modulus for shaft, assumed homogeneous and 
isotropic. 
p=density of material of shaft, supposed uniform. 
o=cross section (and so og=mass per unit length). 
M=mass of load, when there is one. 
I=moment of inertia of c about diameter perpendicular ie 
plane of bending. 

