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Mr. 0. Rod o'er s on the Vibration 



(54) 

 (55) 

 (56) 



The force o£ restitution of the shaft is : 



about GX = ^, 



about G-Y= -—«£", 



about GZ=0. 



If the out-of-balanee couple is ra 2 , the moment of the 

 couple is : 



about GX = — ra 2 sin pat, 



about GtY = to) 2 cos pat, 



about GrZ —nil, 



where p = l or 2. 



The equations of motion are thus : 



— B rj — fcr) + Ao>f = — ra 2 sin pat, 



Bf+/cf + Aft)^ = TCO 2 COS Jt?&)£, . . 



Aai=0; 



or putting D for oj/d£, 



(D 2 + c 2 2 )rj — maD^= rco 2 /B . sin pat, 

 (D 2 -}-e 2 2 )f -f mo)D^ = T&) 2 /B .cospw^ 

 Ai = 0, 



where c 2 2 =^ and m=~-. 



The last equation A&> = gives -co = const., that is, trBre is 

 no fluctuation in the angular velocity of rotation. 

 The other equations give 



{(D 2 + c 2 ) 2 + m 2 a> 2 D 2 } f = ra 2 /B {{c 2 2 -p 2 a 2 )-mpa 2 } co\pat, 



and a similar equation for f in terms of sin pa)£. , 



7. The free vibration is not of importance, as it wil be 



damped out as before, but it is of interest to examira its 



value as illustrating the effect of gyrostatic action oil the 



motion. 



The free vibration is of the form 



x = Ni sin (qt — (f>i) ; y — N 2 cos {qt — cf) 2 ) , 



where N 1? <f>i, N 2 , cf> 2 are constants and 



+ q — ±{ma+ \/m 2 a 2 -\- 4c 2 2 }. 



'7) 



