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IX. On the Vibration and Critical Speeds of Rotors. 

 By C. Ropgeks, O.B.E., B.Sc, B.Eng., M.I.EE* 



NUMEROUS papers have been written on the question 

 of the whirling and vibration of loaded shafts and 

 kindred subjects, and the calculation of the first critical 

 speed — the lowest speed at which the vibration shows a 

 maximum value, is now a matter of daily routine in 

 designing offices. 



This critical speed can be calculated with sufficient accuracy 

 for practical purposes and as a rule the running at speeds 

 not in the neighbourhood of that indicated by the calculation 

 is free from vibration. But cases occasionally arise where 

 troubles from vibration occur at speeds above or below the 

 calculated critical speed, the reason for which is obscure and 

 the remedy correspondingly difficult to find. 



It is the object of this paper to discuss various subsidiary 

 causes which might conceivably lead to unsatisfactory run- 

 ning at other than the usual calculated critical speed, but 

 while these are indicated as possible causes of disturbance, 

 it is not to be assumed that these causes always exist or that 

 they will always induce disturbed running. The object is 

 rather to indicate reasons why vibration might possibly arise 

 and thus if an actual case occurs, to suggest a clue to the 

 cause. 



Although the basis of the paper is a physical or mechanical 

 one, the treatment is largely mathematical, as it is only 

 by this means that formulae can be obtained from which 

 numerical results can be worked out. 



The phenomena when a rotor vibrates are complicated, as 

 the shaft is supported in the bearings on a film of oil, the 

 thickness of which is continually changing, the bearings and 

 foundations are not themselves perfectly rigid, and there is 

 a certain amount of initial bending of the shaft (and to a 

 much smaller extent of the rotor body) due to gravity. If 

 the rotor consists of a number of disks as in a steam turbine, 

 there is also the inter-action of the forces of each disk on 

 the others. 



For the sake of simplicity, we shall confine our attention 

 to the case of a single part rotor, either a disk or a cylinder, 

 rigid as regards bending and mounted on an elastic shaft 

 running in rigid bearings. Some effects of non-rigidity of 

 the rotor and bearings and of alterations in the thickness 

 of the oil film in the bearings will be indicated. 



* Communicated by the Author. 



