﻿and Critical Speeds of Rotors. 131 



3. With the notation given in fig. 3, it will be seen that 

 th^ position of the rotor is completely defined by the co- 

 ordinates x, y, and 6 {x and y being the co-ordinates of" G) 

 and only three equations are required to express the motion 

 fully ; the value of the whirling angle a follows from the 

 magnitude of the other co-ordinates. 



The force exerted by the deflected shaft is err, the com- 

 ponents of which are 



— a{x — e cos 6) along OX and 



— a(y — e sin 6) along OY. 



Resolving along OX and OY and taking moments about 

 G, we thus have : 



M£ + <7(.i'-*cos0) = O, (13) 



My + (r(y-esm0) + Mg=Q, .... (14) 



M^+«ra(tfMn0-yco!«0) = O, . . . (15) 



where k l is the radius of gyration of the rotor about the 

 longitudinal axis through the centre of gravity. 



In practice the rotor is driven at an average angular 

 velocity co say, which will vary from constancy only by 

 small amounts which we shall find later are negligible. 



Assuming as a first approximation that the angular 

 velocity is constant = co, so that 6 — cot, (13) and (14) then 

 become, writing cr/M = c 1 2 as before, 



x + c l 2 x = Ci 2 e cos cot, (16) 



y + c l 2 y = c] 2 e sin cot — g, .... (17) 



while (15) becomes an identity. 

 The solutions are 



■jjrnN^in M-yi)-f — ^-^coscot, . . . (18) 



y = N 2 Cos ('•!* — 71J+ 2 Cl e 2 sin cot-g/c^, . (19) 



C'i — CO 



where N 1; N 2 , and y l are constants. 



4. The above results give the motion of the centre of 

 gravity G. The motion of the centre of the shaft C is to be 

 found in a similar way, still on the assumption that the 



K 2 



