536 Dr. C. Chree on the Whirling and 
The general values of w*, however, in Table 1X. show how 
very eapaly the tendency in M to pr oduce w hirling falls off as 
its position approaches a support, especially when the support 
is such as to fix the direction of the shaft. 
§ 39. In considering liability to whir], we must not lose 
sight of the possibility of the elastic strains and stresses 
exceeding the limit of safety before whirling is reached. As 
an example, let us suppose the shafts circular (solid, or hollow), 
of external radius a and perimeter p (p = 27a), of a material 
for which 
E=20x 108 grammes weight per sq. cm. (1270 tons per 
n = 1/4, sq. inch), 
p =7°8 times the density of water. 
Suppose the shafts unloaded, and denote the maximum 
stress-difference answering to the critical angular velocity of 
whirling by 8, then the following results may be proved :— 


TABLE X, 
Very thin-walled 
Solid Shaft, | hollow Shaft, 
both ends | both ends 


Supported.| Fixed. _Sopportea.| poe 
S (in tons per sq. inch) ... 6-61(p/1)* | 840 (p/ly" 39°7(p/L)}4 | 204-0( p/2)* 
| 








Up | 21:35 1 e208 2011.4 3h 
tta=| 8509 128 Hele | 200 





The value of S for the very thin-walled shaft is really 
6 times that for the solid shaft, for the assumed value 1/4 of 
Poisson’s ratio. Since the maximum stress-difference an- 
swering to the critical velocity varies as the fourth power of 
the ratio borne by the perimeter (or radius) of the shaft to its 
length, it increases with great rapidity as the length is 
reduced, the section remaining unaltered. The physical 
properties assumed answer fairly to steel ; but the results, it 
may be remarked, depend only on the elasticity, not on the 
density of the Tec Under statical conditions, a stress- 
difference of 2 tons on the square inch is but a trifle com- 
pared to what good steel will stand ; but in a rotating shaft, 
where there are ordinarily rapid alternations of stresses from 
various sources, it is probably at least as large a contribution 
from ‘centrifugal forces’’ as a cautious engineer will care 
to see. 
When the ratio of the length to the circumference, or 
