D384 Dr. C. Chree on the Whirling and 
it is easily found from (1) that the frequency answering to 
any arbitrary value of » is given by 
k? = K?—(K?—k,?)o7/@7:" ee 
Thus the critical angular velocity, Q say, being that for 
which & vanishes, is given by 
O? = 0K? /(K? —k,’) aR 
As a check on the applicability of (2), it would in general 
be advisable to determine the frequency k,/27 answering to 
a second angular velucity w,. If (2) is strictly true, we 
should obviously have 3 
(K?—h?for=(K—hjo?, . . . @ 
the quantity on either side of the equation being a value for 
K7/O?; 
In all the cases solved, the quantity vin (1) has approached 
unity as a limiting value when the moment of inertia of the 
load has been indefinitely diminished ; 7. e. the angular 
velocity answering to whirling has approached the limiting 
value 27rn, where n is the number of transverse vibrations of 
the fundamental type executed by the system when not 
rotating in unit of time. 
The feasibility of determining frequencies of vibration 
in actual shaft systems, or models, is a question which I 
must leave to those experienced in Acoustics and practical 
Hngineering. 
§ 37. When a shaft is carried on more than two supports, 
it is not easy to lay down a suitable basis for the comparison 
of the critical velocities answering to different terminal 
conditions. A comparison is, however, easily instituted im 
Cases 1, 2, 4, and 6, when the ‘shaft is unloaded. In all four 
cases, suppose the total length /, the mass m(=apl), the 
ones EI, and let w be the cribieal angular velocity, 
N(=30@/7) the corresponding number of revolutions per 
minute. Then we have the results given in Table VIII. 
TasLe VIII. 



5 moter Value of Value of : 
Cae, Mtaveraiends: w? + (EL /ml*), N-+(EL/mi?)?. 
1. | One fixed in direction, other free. ......... 12-3 aha 33°58 
2. | Both supported .......... Aaa Ra CPOE Eee 97°41 94°25 
4. | One fixed in direction, other supported...) 237-7 147:2 
6. | Both fixed in direction ............cc0c000- 500°6 | 213-7 

