

Transverse Vibrations of Rotating Shafts. 511 
Asa second approximation, neglecting (I')? we find 
wo? = (SHI/MP) (1490 AMP). ...-- (12) 
§ 9. (d) If we know, to start with, that the effect of I’ is 
small, we can simplify the work by taking in place of (9) 
ya Sle? —e fA ae «) (18) 
If we substituted Mgl’/3EI for » we should have the dis- 
placement produced in the shaft by a weight Mg at the end. 
Assuming 7 in (13) proportional to cos &t, and still neglect- 
ing the mass of the shaft, we have 
T=3M (42+ on?) + H1'(y?— 07?) (9/40), 
V=tEI. 377/l 
Thence we have for the frequency equation 
(k? + @?) M+ (k?—@?)(91/4P)=3EL/0, . . (15) 
and for the critical angular velocity 
oe (owl / My (1-9 AM? )-? EG) 
Omitting I’ altogether we deduce (11): while retaining I’, 
but omitting (I’)?, we have (12). 
§ 10. (e) Loaded shaft of appreciable mass. 
Dunkerley’s hypothesis (see § 1) supplies as the equation 
for the critical angular velocity 
ba? Weel /@e, oes) ee Lr 
where w; and 3 are given by (38) and (10) respectively. 
Supposing I’ so small that (12) is applicable, we deduce 
\ opl* ME 311 ; 
agesenr Sul LET > ee 
§ 11. (7) Instead of assuming the truth of (17) we may, 
following Rayleigh, assume (13) as the type of displacement, 
no longer neglecting the mass of the shaft. This adds to the 
value of T in (14) the term 
(14) 
33 
ag Day eat ey 
and so leads to the frequency equation 
ke +0) (M- ee 
( + @°){ M + 140 
For the critical angular velocity, noticing that 140/11= 12°73, 
we have 
opl) + (=o) 9/42 =3EL/P. (19) 
tape, ME ae 
ee Mi’ 3 inet 
Pa ae) Rey Tm a a 


