Shafts in the Neighbourhood of a Whirling Sjieed. 313 



dangerous to run through the critical speed to a higher 

 working speed. 



9. If we include damping we have the displacement of 

 the shaft w = OE, and F = ?n&> 2 . OM; see fig. 1. 



But OE 



\/(g' - m(o 2 y 2 + b 2 a)' 21 



and OM = «xa/-^-. 



V (c -moo 2 ) 2 + 6 J o) 2 



So that we now have after putting k — \/ — as before, 



mao) 2 

 U ~ \/m 2 (k 2 -cD 2 ) 2 ~+b 2 a) 2 ' 



and b=maco-x \/ .. 79 ^ — ^r-^j. 



V m\k 2 — (o 2 ) 2 + b 2 co 2 



It will be noticed that the previous results are a particular 

 case of this. , 



On assuming a particular value for — these results may- 

 be plotted similarly to those in fig. 4. m 



10. Reverting now to the formulae for centrifugal force 

 written above (§ 8) we may compare the forces and vibrations 

 due to a given eccentricity of mass at speeds above and 

 below the whirling speed. 



Consider the design of a shaft for a given duty and 

 operating at a given working speed. 



Then the preceding formulae indicate that it is better from 

 the vibration point of view to design the shaft with its 

 critical speed below the working speed rather than to have a 

 critical speed the same proportion above the working speed. 

 In the latter case the shaft will of course be thicker and 

 somewhat heavier. This result, indeed, is well illustrated in 

 the behaviour of the De Laval steam turbines. 



Thus comparing the centrifugal forces in the two cases in 

 the particular example with which we have been dealing, 

 let F be the centrifugal force when the critical speed k is 

 greater than the working speed co ; and let F' be the 

 centrifugal force when the critical speed k' is less than the 

 working speed ; the eccentricity of mass centre a is supposed 

 to be the same in two cases, but the masses m and m' are 

 not quite the same, the former being somewhat greater 

 owing to the shaft being heavier. 



