(xxii.) 



(xxiii.) 

 (xxiv.) 



520 Mr. S. Lees on the Whirling of an 



or what is the same thing in effect, 



Z(-/ 2 -ar-f 6B/M)-3BL(S)/M =0, 



-3BLZ/I 2 4-e(-/' + o) 2 + 2BL 2 /L 2 ) = 0, 

 -2ricoZp - <l> {r 2 4- I 2 /M)/ = 0. 



By eliminating Z, ©, and <I>, we get an equation of the sixth 

 degree for p. 



The correct method of treating these equations is to solve 

 for p from (xxii.) and (xxiii.), thus getting: — 



I -p 2 -a> 2 +i;B/M, -3BL/M, 



-3BL/I 2 , -jr + co 2 + 2BL 2 /I 2 , 



= 0. (xxv.) 



<E> is then got from (xxiv.), and we see that there is a 

 difference of phase between <£> and Z of tt/2. 



On putting p = in (xxv.), we get the condition for 

 unstable motion ; this is readily seen to lead to (xviii.). We 

 thus arrive at the result that the state of steady motion with 

 the configuration c£ = 7r, is in general stable, bat we get 

 instability (i. e. whirling) when w satisfies (xviii.). 



§ 8. We now proceed to consider the possible oscillations 

 about the steady state of motion defined by (xvi.). 



Taking first the case of motion about <£ = 7r/2, we put 

 cj> = tt/2 4- t/t, where ty is small. Putting z, 0, and ^ pro- 

 portional to e ipi , we get for/?, from (viii.), (ix.), and (x.) ? 



]> 2 + co 2 -6B, M, 3BL/M, r (p 2 + u 2 ) ; 



3LB/T 2 , p 2 -<» 2 -2L 2 B/I 2 , 0, =0. (xxvi.) 



r(p 2 +a> 2 ), 0. yr(r 2 4-I 2 /M). 



This shows that p cannot be zero, in this type of vibration, 

 whatever the values of the constants. The nature of the 

 roots of (xxvi.) can be arrived at by remembering that the 

 equation is a cubic in p 2 , which has positive values for p 2 = co , 



21 2 B 



and p 2 = ; and having negative values for p 2 = co 2 -f — = — , 



and p 2 ——co. Thus p 2 has two positive roots, and one 

 negative root. The existence of the negative value of p 2 

 implies that the motion is unstable. 



If we take the case of motion about <£= - y we are led to 



exactly the same equation lor p 2 . Hence the motion in this 

 case also is unstable. 



§ 9. We have now discussed at some length the motion of 

 the excentrically loaded shaft, neglecting torsional oscil- 

 lations and any yielding at the bearing. It may be remarked 



