﻿134 Mr. C. Rodgers on the Vibration 



10. At the critical speed where co 2 = c\ 2 equations (20) 

 and (21) become : 



x + c 2 x = c'i 2 e cos C]t, 



y + cfy = c 2 e sin erf — g, 



which admit o£ a solution not involving infinite values, 

 namely, 



# = £ — sin c x t 



= *f «*(«.«- |) (26) 



y=—i^ r cosc i t — (?/c l 2 



<\e . / 7r 



= iysini^-- 



)-#i 2 (27) 



Equations (26) and (27) thus give the motion at the first 

 critical speed when friction is ignored. They show that the 

 component along each axis has an amplitude which con- 

 tinually increases in proportion to the duration of the 

 motion, in other words the motion at the critical speed is 

 a spiral of continually increasing radius. 



11. From (24) and (25) it will be seen that the phase 

 difference of the motion with respect to cot changes from 

 zero to 180° as co passes through the value c l} i. <?., as the 

 speed passes through the critical ; (26) and (27) show that 

 at that speed the motion lags behind cot by 90°. It will be 

 seen later that when friction is taken into account, the lag- 

 increases gradually as the speed increases, being still 90° 

 when &> = <?!. 



12. Up to this point we have treated the angular velocity 6 

 as a constant = &), as on the average it will be in practice. 

 Suppose now that it varies slightly from constancy, so that 

 the angular position cat becomes cot + u, where u is so small 

 that its square can be neglected, and we may write smu = u 

 and cos?/ = l. We then have cos 6 = coscot — u sin cot and 

 sin = sin cot+u cos cot, also = u. Substituting these values 



in equations (20), (21), and (15), and writing -^ =c{ 2 as 

 before, we get 



x -f Ci 2 x = w 2 e cos cot— co 2 en sin cot, . . . (28) 



y + c l 2 =y co 2 e sin cot + co 2 eu cos cot— g, , . (29) 



MM + cre(,T sin cot —y cos cot + ant cos cot + yu sin cot) = 0. (30) 



