
Transverse Vibrations of Rotating Shafts. 529 
From Lagrange’s equations I find for the frequency 
equation 
(A? + w”) Mc? (a —c)?(4ab + 4ac—c?)? 
+ (kh? —o’)I’'f4ab(a— 2c) + 3¢(2a? —4ac+ e)}? 
=12HIla?(a+b)(4ab+4ac—c*), . . . . . (21) 
and for the critical velocity 
oe 12Ela?(a +5) 

~ M&(a—c)*(4ab + 4ac—c) a 
fe lV’ {4ab (a — 2e) + 3e(2a° —4ac eOry (22) 
Mc?(a—c)*(4ab + 4ac—e)* 
Neglecting I’ altogether, we obtain (19). 
In the case of two equal spans, or b=a, omitting higher 
powers of I’ in (22), we find 
: 24K Ia* 
————— SAE ooo 
Me? (a —c)?(4a? + 4ac —c’) 
[1 ca —2a?e— 12a ea 



Mc?(a—e)?(4a? + 4ac—e?) 
An identical result is deducible—but not so easily— 
from (18). 
§ 30. Load (M, I’) on massive shaft. 
(7) On Dunkerley’s hypothesis the critical velocity is 
given by 
(23) 
1/w? = 1/@,? + 1/o,’, 
where @, is given by (3) or (4), and w, by (18) with & omitted. 
In the case of equal spans. supposing the effect of I’ small, 
we thus find 
1 _apa* =, : Me*(a—c)? (4a? + Aae—e?*) 
wo 7-41 KI | 24K 1a* 
% I’(4a* —2a?e— 12ac? + 3e*)? 
24H la*(4a? + 4ac—c’) 
(g) The best algebraic type of displacement would probably 
answer to the bending of the shaft under a gravitational 
force supposed to act on both shaft and load, but oppositely 
directed on the two spans. I have only worked out results 
from the simpler type (13), which neglects the influence of 
the load on the displacement. This leads to the frequency 
equation 



(24) 
(?+o°)R+(4—o@")QUV=S8, . . . (25) 
Phil. Mag. 8. 6. Vol. 7. No. 41. May 1904. 20 
