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



Xtl[m f v\lt'(l^-t r )(2-t') 



2(fi + X) 



>riV 2 jZ, {3(1 — 2 -If ]-fiV [mult{l -0(2-0 



du 



-mk^mi-ty-in]- 



These equations determine the deflexions and slope of the 

 curve under the centrifugal loading considered. 



If, now, in turn we make z r and zj coincide with the 

 points of application of the loads, we obtain a series of 



equations connecting w l5 ( — \ , v u ( jzi) > w 2? etc * 



Eliminating these deflexions and slopes from the series of 

 equations we obtain the determinantal equation whose roots 

 are the several whirling speeds. 



§ 16. The case of a uniform shaft freely supported in two 

 end-bearings may be deduced from the results of § 15, by 

 making the shaft of evanescently small diameter in the 

 second span, and the loads on that span also zero. That is, 

 we put ^ = and m! = 0. We then obtain 



^^=(l-^^l^^M^(2-g-^ 2 } + mFj{3^-^(2-g}] 



+ trtlr[mul(l-t){l-3t ) ?-(l-t) 2 } 



+^^{1-3^-3(1-0*}]- 



The determinantal equation for the whirling speeds may 

 thence be developed. These results are similar to those given 

 in 'The Engineer' (pp. 439, 191 (1909)], where the corre- 

 sponding method for the two bearing shaft is described. 



§ 17. As a simple example of the method of § 15, we take 

 the case of two equal loads, one at the centre of each span. 

 Let the moments of inertia of the loads be zero. Let the 

 one span be half the length of the other, and of half its 

 moment of inertia of cross-section. 



Then m 1 = m 1 ! = m; /-c=J, \ = ^ ; k 1 = ki' = 0; t = t' = \. 

 The equations of § 15 then give 



6EI 23 9 



mco-l 6 256 ol2 



12EI 9 23^ 



mcoH^ 1 " 256 Ul+ 512. 



