Transverse Vibrations of Rotating Shafts. 515 
§ 16. Prof. Dunkerley carried out a number of experi- 
ments under the conditions of Case 2. He employed a shaft 
unloaded, or loaded with one or other of two pulleys of dif- 
ferent sizes. As a preliminary to comparing his theory with 
observation, he had to calculate critical speeds for the pulleys 
alone, neglecting the mass of the shaft. Table I. compares 
the number of revolutions per minute which he calculated 
from a formula equivalent to (8), with & omitted, with the 
corresponding numbers which are given by the much simpler 
formule (9) and (13). The results apply of course only to 
the particular shaft and pulleys employed by Dunkerley. 
Tasie I.—N, (Number of revolutions per minute for 






whirling). 
) gee | ia a ee ie (Se Pare he 
Pulley. Formula. O/J=| 1/2. |; 1/3. | 14. TAA: 1/32. 
; = es ——— ———————s a 
(8) [Dunkerley’s].... 1495 | 16838 | 2705 4621 13537 
is (9) | 1495 | 1682 | 2690 4532 12343 
(13) | 1495 | .... | 2710 | 4636 | (163864) 
«1 (8) [Dunkerley’s]...;- 997 | 1122 | 1808 | 3116 | 10355 
coe (9) | 997 | 1121 | 1794 | 3015 |} 8231 
PGS) aes iaal Sea 1808 3119 (18716) 
| | | 

§ 17. The results from the formula (13) for the smallest 
value of b// are put in brackets because the assumption on 
which the formula is based—viz., that the contribution from 
I’ is small—is then far from being satisfied, so that the result 
is @ priori unsatistactory. The closeness with which the 
simplest formula (9) approaches to the results from the com- 
plicated formula (8) is rather surprising. 
In Dunkerley’s case the value given by (3) for N,—the 
critical number of revolutions per minute for the shaft when 
unloaded—is 1122. Thus for the value 1/32 of b/l we have 
@,/@,= N./N,=13537/1122 for pulley I. (or 12 roughly), 
= 10355/1122 a OL tor UO roushly): 
Thus the contribution from @,, or Ny, to the critical speed 
for the loaded shaft on Dunkerley’s hypothesis (1} § 1 is only 
about one-eightieth of that from @,, even for the case of the 
heavier pulley II. Under such circumstances an error of 
even 100 per cent. in the value of 2, or Nz, would exert but 
little influence. It is thus clear that under the conditions of 
Dunkerley’s experiments the simple formula (9) would for 
all practical purposes be as satisfactory as (8). 
§ 18. Table Il. compares the speeds at which Dunkerley 
observed whirling to commence with those which he 
2N2 
