636 Prof. H. H. Jeffcott on the Whirling Speeds of a 



the shaft diameter in many cases cannot be made very large 

 and stiff, and in consequence, the working speed exceeds the 

 lowest whirling speed. No real disadvantage, however, 

 results from this circumstance provided the rotor be properly 

 balanced, as the machines can be run easily and quickly 

 through the whirling !«peed. Numerous electric generators, 

 de Laval turbines, etc., are run through the lowest whirling 

 speed every day without adverse result. 



Moreover, in the case of some turbo-generators and 

 blowers, the machine is often in operation day and night for 

 long continuous periods, so that it is only occasionally that 

 they have to be run through the whirling speed. Difficulties 

 do not arise when the rotors are well balanced, and the- 

 advautages that result are worth securing. 



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It will be gathered from the foregoing that it is a problem 



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of practical importance to determine the values of the first 

 and second whirling speeds of three-bearing rotors as well 

 as of two-bearing shafts. 



A loaded shaft is said to whirl when it rotates in a deflected 

 form, the centrifugal forces and moments of the loads being 

 equilibrated by the elastic restoring forces of the shaft. 

 This condition can take place when the speed of rotation 

 approximates to a speed of free transverse vibration of the 

 loaded shaft. If the speed is maintained constant at such a 

 value the bending of the shaft may become excessive and 

 damage may ensue, unless the restraining action of the 

 bearings, which are then subjected to severe alternating 

 forces, is sufficient to prevent excessive deflexion of the 

 shaft, 



Various methods for the solution of the problem of the 

 whirling of a loaded shaft supported in two bearings have 

 been published, more usually confined to the determination 

 of the first whirling speed only. Prof. Morley has shown 

 how the graphical method of solution determining the dis- 

 placements may be used with successive approximations for 

 the determination of the first whirling speed of a loaded 

 shaft supported in three bearings ('Engineering/' vol. cvL 

 pp. 573, 601 (1918)). But the graphical method does not 

 appear to lend itself to the determination of the second and 

 higher whirling speeds. 



It is proposed in the following pages to deal analytically 

 with the first and higher whirling speeds of a shaft supported 

 in three bearings, loaded in any manner, and of sections 

 varying from place to nlace along the length. The various 



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speeds may be obtained by equating the action of the eentri- 



