﻿and Critical Speeds of Rotors. 151 



the centre of gravity, are A and B respectively, the angular 

 momenta are *: 



about GX, ^B^-SHAwf, 



about GY, ft 2 =B(?f-ff) + A©?7, 



about GZ, A 3 = B(^-^) + Ao)f. 



In the actual case the angle yjr is very small, so that we 

 can put sin-v/r = ^ cosi/r=l, and f=l, f=0j also the 

 products f 77 and 77^ are both negligibly small. 



We thus get : 



7*i =— B?7 + A&)f, 



7l 2 =:B| + Aft)>7, 



7i 3 = Ao). 



* Another and perhaps more legitimate way of deducing these 

 equations is as follows : — 



If a lt wo, and o) 3 are the instantaneous angular velocities about moving 

 axes 3rX', GY', and GL fixed in the rotor and moving with it : consider 

 the iistant when GX' is perpendicular to GL and GZ (cf. fig. 5), and 

 let )e the angle between the planes LGZ and YGZ. 



Th«nw l =— -^ w 2 = 0sin^, and w, = w. 



Th( angular momenta about GX', GY', and GZ are Bo>i, Bw 2 , and 

 Aw. The anuular momenta about the fixed axes GX, GY, and 

 GZ aie : 



Ai = B{ w l cos 0+w 2 cos \p sin 0f4 Aw 3 shuf/sin 9, 



h 2 = B { — o>i. sin 9 4- w 2 cos \p cos 9 \ -j- Aw 3 sin ^ cos 0, 



7i n — B{ — w 2 sin^}+Aw 3 cos^ ; 



that i; 



hi= B{— ^ cos 9-\-9 sin ip cos 4> sin 0} + Aw 3 sin;// sin 9, 



h 2 = B{ -4/ sin 0+6/sin^cos4/cos0} + Aw 3 sin^cos0, 



h = — B0 sin 2 iff+Awj cos 1//. 



Al.o 



S = sin^sin0 and £=ij/cosi//sin 0+0 sin ^cos 9, 



J7 = sin'4/cos0 and r/ = ^ cost// cos 0—0 sin ^ sin 9, 



£ = cosip. and £== — i£sin^; 



so thit 



tlK—Ki = —yf cos 0+0 sin cos^sin0. 



Zi-ZZ— i\j sin 04-6* sin ^ cos $ cos 0. 



% n — ■,)'%= — O&m 2 ^. 



ty substituting these values in the equations for A t , A 2 , and A 3 , we 

 obain the relations given above. 



