

Transverse Vibrations of Rotating Shafts. d09 
~ 1’=moment of inertia of M about an axis through its C.G. 
perpendicular to plane of bending. 
w=angular velocity of rotation. 
k/2a7=frequency of lateral vibration, taking rotation into 
account. 
_ T=variable part of kinetic energy of system. 
= ,, potential ,, i 
p=0'(po/EI)}. 
Case 1 (Dunkerley’s Cases I., VIT., and 1X.). 
§ 6. Overhanging shaft fixed in direction at end A. 
Length AB=1. 
' The terminal conditions are : 
Ab Anjar.2—0), .y=dy/dr—0 ; 
edi pean —!). - da gide*=d'y/dx* =(. 
_ (a) Unloaded bar : Euler-Bernoulli solution (¢7. Dunkerley, 
lsc. p. 288). ; 
. The displacement is of the type 
y=a$ (cosh wx—cos px) (sinh pl +sin pl) 
— (sinh wx—sin ux) (cosh wl+cos pl)},.. (1) 
where a is a constant. 
The equation for s—and so for w—(see Dunkerley’s 
equation (A) p. 289) is 
1+cosh p/ cos pl=0. ed, ean) 
. The smallest root (see Rayleigh’s ‘Sound, art. 174) is 
pl =1-8751. 
But by definition of py, 
wo? = (BI /opl!)(ul)$, 
‘ode. wo? =12°36(El/apl), . . . . . (3) 
| Oro =F peep ys PL 
_ Usually I shall record only the value of w?, as often more 
convenient than that of w. In general w? will equal (EI/cp/') 
multiplied by some numerical quantity. 
The above method leads to a somewhat complicated ex- 
pression for the displacement, and throws no direct light on 
the relationship of whirling to lateral vibrations. 
_§ 7. (4) Supposing the bar still unloaded, replace the 
Euler-Bernoulli expression (1) by the much simpler one 
y= (x*—A4 lx’ + 6 [Px*), gaara (5) 
