344 REPORTS ON THE STATE OF SCIENCE, ETC. 
The free transverse vibrations of a disc of uniform thickness, entirely uncon- 
strained, were investigated by Kirchhoff in 1859.6 The normal modes are 
characterised by concentric nodal circles and by equally spaced nodal diameters : 
for any given number (s) of nodal diameters, the frequency rises with the 
number (n) of the nodal circles, and in regard to the higher frequencies it is 
found that the effect of increasing n by unity is approximately the same as that 
of increasing s by two. The frequency of the free vibration depends to a slight 
extent upon the value of Poisson’s Ratio, of which a representative value for 
steel is 0.3. 
Kirchhoff’s solution fails in respect of those types of vibration in which the 
number of nodal diameters is unity or zero, when the disc is attached at its 
centre to a shaft and rotor of inertia sufficient to prevent any change in the 
position or slope of the disc at that point. The necessary modifications have 
now been investigated, and the graver frequencies of vibration calculated for all 
types of vibration possible to a non-rotating disc.’ It is found that substantially 
exact values could have been obtained with much simpler analysis, by using.a 
method due to the late Lord Rayleigh, in which the type of the deflection 
occurring in the vibration is assumed, and an estimate of the frequency derived 
from the corresponding expressions for the potentiai and kinetic energies. The 
method is particularly valuable, in that it may be applied with equal accuracy 
(assuming the validity of the theory of thin plates) to discs of curved profile : we 
are thus in a position to calculate the graver frequencies of vibration for any 
given disc. 
The effects of rotation, in a dise of uniform thickness, were discussed by 
Prof. Lamb and the present author in the paper to which reference has been 
made above. When the disc is stressed by centrifugal forces, its effective 
flexural rigidity is, as we have seen, increased : even though it were practically 
a membrane, without any flexural rigidity of its own, it would vibrate with 
finite periods under the restoring effect of the centrifugal system. The problem 
first considered, therefore, was that of finding the frequencies of free transverse 
vibration when the centrifugal system acts alone, and this was solved without 
difficulty in the case of a uniform disc. ‘lhe modes are generally similar in their 
nature to those of the non-rotating disc, and the frequency, in any given mode, 
varies directly as the speed of rotation. No alteration in frequency is entailed 
by constraints acting at the centre. 
In the general case, where vibration occurs under a restoring system to 
which both the flexural rigidity and the centrifugal stresses contribute, an exact 
solution would be very difficult to obtain, principally for the reason that the 
type of the vibration will itself change with w. But the need for an exact 
solution is obviated by Lord Rayleigh’s theorem, that a small error in the 
assumed type of vibration, when the frequency is estimated by his method, leads 
to an error in the frequency which will be of the second order; in fact, there is 
no need to do any further calculation whatever, since it can be shown that the 
gravest frequency corresponding to any definite number of nodal diameters will 
be given, with close approximation, by an expression of the form 
p’=pi+p.", 
where p, is the frequency found by neglecting the rotation, and p, is the 
frequency found by neglecting the flexural rigidity. 
It has been shown, further, that the gravest frequency will be underestimated 
by the formula just given. This result suggests a very simple method for investi- 
gating the gravest frequencies natural to a disc of curved profile : for the method 
of Lord Rayleigh may be applied with equal accuracy to the calculation both of 
p,? and of p,?, and in each case, if the gravest frequency is under investigation, 
it will give figures for these quantities which are either exact or over-estimated. 
Thus the two errors involved in these approximate methods tend to cancel one 
another as regards their effect on the estimated value of p?, and a closely 
approximate result may be expected. 
6 “Ueber die Schwingungen einer elastischen Scheibe,’ Crelle’s Journal, 
vol. 40 (1850), and Pogg. Ann., vol. 81 (1850). The essentials of the analysis are 
reproduced in Lord Rayleigh’s 7'heory of Sound, §§218, 219. 
7 It is these graver frequencies which are of primary importance, since 
vibrations of high frequency will in practice be eliminated by air damping, etc. 
