Transverse Vibrations of Rotating Shafts. a0 
The smaller w, or N, the more easily is the shaft caused to 
whirl. Table VIII. thus serves to bring out the great re- 
duction in the tendency to whirl caused by fixing the direction 
at the ends of the shaft. | 
§ 38. A comparison is also readily made for a loaded but mass- 
less shaft under the conditions of cases 1, 2, 4, and 6, when the 
moment I’ of the load is negligible. Thisis done in Table IX. 
The load M is supposed at the end in case 1; in the other 
cases its distances from the ends A and B are aand 6. The 
so-called “ minimum values” of w? and N answer to that 
position of M which vives the smallest critical velocity. 
The rest of the notation has the same significance as in 
Table VIII. 
TABLE IX. 

General value of Value of b/a | Minimum Values of 
Case. | End A. End B. | w?+(EI/M@). giving minimum|— 










value of w. — |w?+(EI/M/3)|N+(EI/MBy 
1. ...| Direction | Free. Sea Mba) Pov Sec obe | 3 16°54 
fixed. 
2. ...| Supported.) Supported.) 31/a2b> 1 48 66°16 
ae 3 is 9)3 
eee — | Supported. 12 Gat b) (1/2) | 101-9 96-4 
6. ...| Direction | Direction 318/33 1 |  192:0 132:3 
fixed. | fixed. | 


In cases 2, 4, and 6, the position of the load which supplies 
the minimum value is precisely that for which Dunkerley’s 
equations and mine agree in making the effect of I’ vanish. 
Again, in every case where J have treated I’ as small, the 
corrective term in I’ has increased w. There are thus reasons 
for regarding the above minimum values of N as pretty safe 
measures of the lowest speed at which whirling is likely to 
arise when a load is attached anywhere to a shaft of much 
smaller mass. 
If we compare the results for w? or N in Table VIII. with 
the corresponding minimum values in Table [X., we obtain the 
following as the ratios of m to M for which the critical 
velocities in the two cases are equal :— 
Case. oo 54, 1. 2. 4, 6. 
ref DA 3 n.t 30 4°12 2°03 9-30 2°61 
It follows that when a load is near its most effective position, 
it must be taken into account as well as the shaft if we aim 
at a close approximation to the critical velocity. 
