Transverse Vibrations of Rotating Shafts. 519 
In the statical problem »=gap/24EI. In the kinetic 
problem, assuming 9 & cos kt, we find 
(K-40) | 9Met(e + 1)*(Be2 + el P+ op | P— TP) 
+ 9 se —@)?+3EP(4EP—P)? + bat cl(4e—P) + tbe @ } | 
oe ) a 
+ (Me) x 9f4ee(I-+0) Cy = "PEL (PPE + 101 + 6%). (1M 
Putting k=0, we have a second formula for the critical 
velocity, in which allowance is made for both shaft and 
load. 
If in (14) we omit M and I’ we have a result appropriate 
to the unloaded bar in (a). It is not, however, very satis- 
factory unless c/J is small. When powers of ( (</) above the 
fourth are neglected it gives for the critical angular velocity 
@? =97-55(El/opl!){1 —0°06(c/l)?— 6°8(¢/2)? + 3°5(e/l)*t, (15) 
a result very similar to (3). 
(g) An assumption which appears more natural at first 
sight than that made in either (e) or (f) is that the type of 
displacement answers to the bending of the shaft under its 
own weight and that of the load combined. This gives 
1 
for CB, y=n4 ge eel ue Cae: nh, ie 
‘aP : & 3E/\ 
pee = 4s BE he 5 a0 ere) Yh 
where &=c—ua, Pl= Me—top(?—@), a7 
&= |—a', H=4(PP4Me) 4 gon(h+0).5 ) 
Writing for shortness 
i . 2 917 9 7 EE 2 
R=MCH? +p | gas (P" + Me!) + p69 cP (PE + Me’) + = (+0) | 
eae 
+30H?— eM H— en: 
f 
Q=(H +5 eM + 57 eap)’, 
3 313 : 1 were 
S=EI(i(P*l? + Me!) + fop(P-+ Met) + 55 (op)*(P +0}, 
