Transverse Vibrations of Rotating Shafts. d13 
equations (see Appendix, § 41) we find for the frequency 
equation 
{M(?+ @?) —38 El (a? +0-%) } {/(—o?) —3EI (a7 £0-1)} 
= 9(EI)?(b-?—a-?)?. (8) 
From this, putting £=0, we obtain a quadratic equation for 
the critical angular velocity, identical with that found other- 
wise by Dunkerley (/. c. p. 307). 
If in (8) we absolutely neglect I’, we have the simple 
result 
ewer — oll t/Marb* s :.. aieauren) Co) 
whilst, retaining I’ but neglecting 1%, we have for the 
critical angular velocity 
/ — 
o'= (SEI Ma*bs) 4 1+ : (“ : 
2 
— ) }. 0) 
(d) Tf in the last sub-case we start with the assumption 
that I’ is small, we may take 
for AC, y =nbzx (P—L?—2?), 
for BO, y’=nae!(2—a2—2’) )- 
lf » were replaced by gM/6EI/ this would represent the 
bending of the bar under the weight of M. 
Assuming 9 « cos kt we find by Lagrange’s equations 
(k? + w?) Mab? + (k?— w*)I’'(a —6)?=3EIl. . (12) 
(11) 
For the critical angular velocity, putting =0, we have 
w? = (3EII/Ma2b?){1—(I//M)(a—b)?/a?b7}-1.. (1) 
When (1’/M)? is neglected this is identical with (10). 
§ 14. (e) Loaded shaft of appreciable mass. 
On Dunkerley’s hypothesis we have for the critical velocity 
Le*= 1/w? + 1/03, 
where 7 is given by (3), and w2 by (8) with & put =0. 
Thus when I’ is so small that (10) is applicable, we have 
aie 2.) ape Mat?  I'(a—b)? 4 
o 9741E1  3EHi/ 3H ‘nay 
(f) If we employ the same Rayleigh formula as in (c), 
Phil. Mag. 8. 6. Vol. 7. No. 41. May 1904. 2N 

