

Transverse Vibrations of Rotating Shafts. 507 
inertia) of the load requires to be taken into account, it may 
be shown that if 4/27 be the frequency of the vibrations 
which the shaft executes when displaced laterally at a time 
when it is rotating with uniform angular velocity o, and K/27 
be the corresponding frequency in the absence of rotation, 
then 
Fe Rae nt oe (ay 
As @ is increased, the frequency of vibration and so the 
stability of the bar aiyanencie until eventually when 
the frequency becomes nil, i. e. the period becomes indnite, 
or the righting power vanishes. In fact, the position is 
similar to that of a ship whose C.G. has come to coincide with 
the metacentre. The case is not one in which forced vibra- 
tions are set up with a frequency equal to one of the natural 
periods. What leads to whirling is the indirect action of the 
rotation in reducing to zero the rightin g forces which naturally 
act on the shaft when displaced laterally. The case of a 
loaded but massless shatt is of course an extreme one: butall 
the other cases which i have examined present similar features. 
The case selected for mathematical treatment in the Appendix 
is that of a shaft supported at both ends; this admits of a 
variety of sub-cases, illustrative of various points. 
| March 15.—To prevent misconception, it seems desirable 
to state explicitly and prove—as was done when the paper 
was read—that the formula of § 3, 
k? + w? = K?, 
apples exactly io all unloaded shafts, to the degree of 
accuracy possessed by ordinary equations for thin rods. 
The elastic bodily equation has the followi ing forms : 
*y/dx* = ©? (opy/KI) for rotation with whirling velocity O, 
= K*(opy/E1) for vibration without rotation, 
=(k° + @*)(opy/E1) for vibration when velocity of 
rotation is w. 
The value of w in the typical equation 
d*y/dz' — pty 
depends only on the terminal conditions. Thus for any, the 
‘same, system of supports we have 
i? +o? =K? = 0°.) 
