522 Whirling of an Eccentrically -loaded Overhung Shajt. 



Disregarding the last of these for reasons which have 

 . . . • 



been indicated earlier, and putting the variables proportional 



to e ipt . we oet 



-/r-ar-H)B/M, -p 2 -co 2 , -3BL/M, 

 — p 2 — or, —p 2 — to 1 + Kj , 0, 

 -3BL/L, 0, -p 2 -f co 2 + 2L 3 B/I 3 



0, 



0, 



-p 2 + a> 2 , 



K 2 , -/.) 2 + ft> 2 , 



This may be expressed in the form : — 

 6B/M, -p 2 -o) 2 , -3BL/M, 



-Kj-GB/M, K l5 -K 2 +3BL/M. 

 -3BL/I 2 , 0, 2L 2 B/I 2 , 



= 0. 



-;> 2 +a> 2 + K 3 ,, 

 . . (xxxiii.) 



0, 

 K 2 , 



0, 



-K a + 3BL/I 2 , K 2 , 



0, 



-K 3 -2L 2 B/L, -jt? 2 + co 2 + K 3 , 



. . . (xxxiv.) 

 a quadratic equation in p 2 . 



It should be noted that in (xxviii.) to (xxxi.), the values 

 of z, z lt 0, and 1 are not the absolute values, but give the 

 oscillatory values about positions of steady motion defined 

 by:— 



-rco 2 - ( o 2 (z-{-z ] )+? ) B(2z-L0)/M = Or\ 



-^(r + rJ + Kn + KA =0, I 



*> 2 (0 + £ 1 ) + B(-3*L + 2L 2 6>)/I 2 =0, | 



g>X0 + 0O + K 2 z 1 + 'K.30 1 =0,j 



These may be solved for c, z l9 0, and lt in the usual 

 manner, by determinants. In particular, if 



A = 



tiB 



~^/r CO' 



M 



— Q)- 



3BL/M, 0, 



2 , Ki-o) 2 , 0, 

 2BL 2 



-3BL/I 2 , 0, 



0, K 2 



L 



l 9 



K 2 , 



+ K 3 , 



= 0, (xxxvi.) 



the values so obtained become infinite, and we get whirling. 

 It will be found that this last equation is a quadratic in o> 2 ; 

 in fact, is identical with (xxxiv.), when p is put equal to zero. 

 § 11. The treatment we have given would not be accurately 

 applicable if very high frequency vibrations were considered; 



