Le OAs Al 
LX. The Whirling and Transverse Vibr cations of Rotating 
Shafts. By C. Cures, Se.D., LLD., FRS= 
CONTENTS. ‘sta 
~§§ 1-5. Preliminary Discussion. 
6-11. Casr 1. Overhanging shaft, end fixed in direction. 
12-18. Casz 2. Shaft “ supported ” at both ends. 
19-25. Casn 3. Ov erhanging shaft ‘ supported ” at one end antl 
at a second point. 4 
26-27. Case 4. Shaft “supported” at one end, and w ith. 2 
direction fixed at the other end. 
28-32. Case 5. Shaft “supported ” at both ends and at inter: 
mediate point. . 
33-35. Casn 6. Shaft fixed in direction at both ends. i 
36-39. General Conclusions. . 
40-46. Mathematical Appendix. ee 
Pr eliminary Discussion. 
§ 1. ATERAL vibrations are those executed by a bar 
when bent to one side and then released. They are 
connected with the “whirling” of rotating shafts, but ina 
way which has not, I think, hitherto been clearly recognized. 
The subject of “ whirling ” has been treated by Professor 
Greenhill fin a well- on paper dealing with an unloaded 
shaft rotating under various terminal “conditions. More 
recently the subject has been treated in an elaborate and 
important paper by Prof. Dunkerley t. He employed two 
ways of calculating the critical speeds of rotation. The first 
makes use of the ordinar vy elastic solid equations applicable to 
thin rods acted on by “ centrifugal force’; this is the method 
followed by Greenlill. On attempting to apply this method 
to loaded shafts, Dunkerley reached results which he consi- 
dered hopelessly complicated. In his second method, which 
is due apparently to Prof. Osborne Reynolds, Dunkerley 
calculated a critical speed for the loaded shaft, in which the 
mass of the shaft itself was neglected. Calling the frequency 
thus obtained N,, and that found for the unloaded shaft by the 
first method N,, he deduced a final value N for the frequency 
from the equation : 
1/N?=1/N74 T/ NS. 0 ae) 
By speed or frequency Dunkerley means the number of 
revolutions per ibrbias: Iie: 
=30@/ 7 6. 2 (2) 
where o is the angle thr eh which the shatt rotates in one 
second. 
* Communicated by the Physical Society: read March 11, 1904, 
+ Proc. Institution of ern Engineers, ee Be 162. 
{ Phil. Trans. A, 1894, p. 27 
