Excentrically-loaded Overhung Shaft, 519 



in the last three equations. In this way we obtain : — 



7ft) 2 cos(/>-w 2 c + 3B(2c-L(9)/M = 0, . . (xi.) 

 W# + B(-3^L + 2L 2 0)/I 2 ==O, . . (xii.) 

 — rza)' 2 sin 2 (f> = Q. . . (xiii.) 

 From (xiii), we see at once that either 



z=0 (xiv.) 



or sin = 0, which gives 



<£ = or 7r ( xv -) 



These values we now proceed to discuss. From (xiv.), 

 (xii.), and (xi ), we find that it c = 0, then 6 = y whilst 

 cos = 0, giving 



, tr 3tt . . . 



0=2 or ~2~" '" ' * ' ' ( xvl *) 



Taking (xv.), (xii.), and (xi.), we find two equations for 

 z and 6. Thus for z we get 



-■j(6B/M-a) 2 )(2L 2 B/L + a) 2 )-DL 2 B 2 /MI^ 



±™ 2 (2L 2 B/I 2 + s> 2 =0. (xvii.) 



There will be two corresponding values for 6. If we take 

 = 0, we get z and 6 negative. We therefore take = tt as 

 the value in (xv.), leading to a real steady value of z. The 

 values of z and 6 will be infinite if 



(6B/M- - ft) 2 )(2L 2 B/I 2 4-ft) 2 )-9L 2 B 2 /MI 2 =0. (xviiij) 



This is a quadratic equation for co 2 , and it will be found to 

 be identical with the result given by Ohree (loc. cit. eq. 10), 

 for the whirling speed of an overhung shaft with a sym- 

 metrical flywheel load. Other things being equal, the steady 

 values of z and 6 are seen from (xvii.) to be proportional 

 to r, as might have been expected. 



§ 7. To examine possible oscillations about a state of 

 steady motion for the case of <f> = 7r, we put r = c + Z, 

 = Q + ®, = O -f<£>, and substitute in (viii.), (ix.), and (x,). 

 As we are to put Z=Ze ipt , = 0^', ® = <S>e ip \ with Z, 0, 

 and <£ small, we neglect terms of the second and higher 

 orders in Z, 0, and <J>, when substituting in (viii.), (ix.), 

 and (x.). In this way we are led to 



Z-o> 2 Z-k3B(2Z-L0)/M =0. . . (xix.) 

 + a> 2 + B(-3ZL + 2L 2 0)/I 2 = O, . . (xx.) 

 r 2 4> - 2rZo> + I 2 $/M =0, . . (xxi.) 



