
506 Dr. ©. Chree on the Whirling and 
Dunkerley’s second method, in so far as it relates to finding 
the critical speed for a loaded but massless shaft, is really 
analogous to a method illustrated by Lord Rayleigh * in 
obtaining approximate frequencies of vibration. Beth 
assume the displacement of the bar to be of a simple 
algebraic type; but Dunkerley applies ordinary statical 
equations, whilst Rayleigh applies dynamical equations de- 
duced by means of the principle of energy. Rayleigh, how- 
ever, advances in justification of his method a result based on 
very general reasoning, viz. that a considerable departure 
from the true type of vibration leads to only a small error in 
the estimate of the frequency. 
I am not prepared to say that Rayleigh’s general theory is 
impervious to criticism. A general theorem may pass muster 
even with acute critics, simply from failing to suggest points 
of view which decline to be left out of account in actual 
practice. Again, a theorem may be practically satisfactory 
within certain limits, and yet those limits may be so difficult 
to recognize that applications may be fraught with peril to 
any but one of the very few men who combine profound 
physical insight with first-rate mathematical ability. Still, 
taking all these things into account, I think it will be generally 
recog nized that, in view of the empirical nature of Dunkerley’s 
second method, ‘the application of Rayleigh’s method to the 
problem of whirling is, if practicable, highly desirable. 
Numerous applications of it will be made here, and there is 
an illustration of the mathematical details in the Appendix. 
§ 3. Before treating individual cases, I shall describe in 
unmathematical language the true nature of the connexion 
between lateral vibrations and the phenomenon of whirling. 
Ordinarily, when a shaft held at one or both ends is acted on 
by forces tending to bend it, on the removal of these torces 
it tends to return to its original straight position ; in doing so 
it overshoots the mark and vibrates to and fro laterally. The 
velocity of its approach to the equilibrium position, and the 
frequency of the vibrations subsequently executed, are greater 
the larger the elastic stresses produced in the bar by a given 
lateral “displacement. When the bar is rotating round its 
longitudinal axis, and is displaced laterally, the elastic stresses 
tend as before to bring it back to the undisturbed position ; 
but the “centrifugal forces” have exactly the opposite 
tendency: they thus reduce the righting forces, and so 
diminish the fr equency of vibration. If we take the simplest 
case where there are no complications from the mass of the 
shaft itself, and where only the mass (not the moment of 
* “Theory of Sound,’ vol. i. Arts. 182, 183, &e. 
Pp) ‘“ 
