— = 
, 
a 
‘a 
Vibrations of Rotating Shafts. — 505 
When more than one load exists, Dunkerley calculated 
critical speeds Ns, N3, ... for each separately, and deduced 
a critical speed for the whole system from the formula 
BNF SS 1 NENG ep ees Pe (8) 
He made a number of experiments witha miniature shaft, 
loaded with one or both of two pulleys, and in many cases 
the speeds at which whirling commenced agreed remarkably 
with those calculated. In some cases better agreement was 
obtained with a formula of the type 
Ne 1yNs? =: ajN*; 
in which a is an experimental constant. 
Dunkerley (/. c. p. 281) noticed that there was a connexion 
between the speed of whirling and the frequency of the lateral 
vibrations of the shaft when not rotating ; but I see no indi- 
eation in his paper that he had grasped the real nature of the 
connexion. 
Dunkerley’s method (/. ¢. p. 358) of arriving at his working 
equation (3)—which he characterizes himself as “empirical” — 
is not convincing mathematically. In fact, the result does not 
appear to bein general strictly true ; and there seems nothing 
in Dunkerley’s work which serves to bring out the limitations. 
As we shail see later in particular cases, the agreement 
between theory and observation is not by itself sufficient 
evidence of the general applicability of formula(1). Again, 
the applications by Dunkerley of the Huler-Bernoulli elastic 
theory are cumbrous mathematically, and the formule to 
which they lead, and which were employed by Dunkerley, 
often admit of great simplification, without appreciable dimi- 
nution of accuracy, under his experimental conditions. Also, 
as already explained, the true nature of the relationship to 
lateral vibrations is not brought out. 
For these several reasons I have thought it worth while to 
examine the whole question afresh from a variety of points of 
view, making liberal use, however, of Dunkerley’ S experi- 
mental results, and referring to his formule in the several 
cases, so that ‘the present work is in many respects supple- 
mentary to his. To go fully into the mathematical inves- 
tigations in each case would occupy an undue amount of 
space, and as the same methods are employed in the different 
cases, I have deemed it sufficient to give one or two illustrations 
in the Appendix at the end of the paper. 
2. It is unnecessary to describe the method employed by 
Greenhill and Dunkerley for the unloaded shaft, as it is 
simply the approximate Bernoulli-Eulerian method de- 
scribed in mathematical textbooks treating of “thin” rods. 
