COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 343 
Problem IT. Caleulation of the Transverse Deflections Indueed in a Dise by 
a Specified Load System, 
Two cases of this problem arise :— 
(a) when the disc is stationary ; 
(6b) when the disc is rotating. 
It is evident that rotation must increase the effective flexural rigidity of the 
disc, since work has to be done against the centrifugal stresses by any agency 
which produces transverse deflections. ' 
Under the heading (a), we have exact solutions for the deflection of a dise of 
uniform thickness (type A), under various systems of transverse load which are 
symmetrical about its axis.° Recent investigation has shown that the analysis 
tan be extended without difficulty to any system of transverse loading which 
varies in intensity as some power of the radial distance, and that a further 
extension may be made to discs of type B. Further, in this problem also it 
appears that graphical methods can be employed to find the deflection due to any 
joad system (axial symmetry only being postulated), in a dise of any specified 
profile : but here again it seems doubtful whether any real demand exists for 
methods of such generality. 
Under the heading (6), we have to consider the effect of centrifugal force, 
and the difficulty confronts us that the form of the deflected disc, as well as the 
absolute magnitude of the deflections, will in general be altered by the rotation : 
had a change in magnitude been the only effect, we could very easily have deter- 
mined its amount. But since the centrifugal forces will tend to reduce the 
deflections, the exact magnitude of their effect is of less practical interest than 
would have been the case if it had increased the danger of rubbing between 
fixed and moving parts, and in view of the analytical difticulties associated with 
an exact solution, it will probably be sufficiently accurate to start with the exact 
solution for a non-rotating disc; to calculate the additional transverse loading 
which would be required to maintain this deflection against the centrifugal 
stresses corresponding to the specified speed of rotation; and to estimate a 
mean coefficient whereby this may be represented as a fractional addition to the 
original loading : we shall then have an idea of the centrifugal effect, expressed 
roughly in the form of a fractional addition to the flexural rigidity. 
Problem III. Calculation of Critical Speeds (or of the Normal Frequencies 
of Free Transverse Vibration) in a Dise of Specified Profile, Stationary or 
Rotating. 
There is evidence that resonance is liable to occur between the frequencies of 
free vibration which are natural to the turbine disc and the frequencies of the 
small periodic disturbing forces which come into play when the turbine is 
running. The designer’s problem is to arrange that such resonance shall not 
occur, at all events within the practical speed range, and evidently there are, 
theoretically speaking, two ways of effecting this result : either he must eliminate 
the periodic disturbing forces, or he must arrange that their frequencies shall be 
less than any of the frequencies natural to the disc. In practice, the disturbing 
forces are not always avoidable, and their origin is sometimes obscure : but 
their nature can best be guessed from the nature and frequency of the vibration 
which they excite, and thus our most important problem is the determination of 
the natural modes and frequencies of free transverse vibration, for a disc of any 
specified form. ; 
‘As in problem II, a distinction must be drawn between the effects of the 
flexural rigidity and of the effective rigidity induced by rotation, and the neces- 
sity for taking rotation into account is much greater in the present instance : but 
it fortunately happens that the requisite allowance for it can be made much 
more easily. From experience gained in similar problems we may expect that the 
nature of the vibrations will be different according as the disc is entirely free 
or to some extent constrained, and that the former conditions will yield the 
simpler solutions. ; 
° Cf. A. E. H. Love, Theory of Hlasticity, §314 (a); and A. Morley, 
Strength of Materials, chap Xili. Sip? Ha 
