Transverse Vibrations of Rotating Shajts. 525 
This agrees with (12) when I’ is neglected, with (13) when 
(1’)? only is neglected. 
§ 27. (e) Load (M, I’) on massive shaft. 
On Dunkerley’s hypothesis we have 
1/o?= 1/0? + 1/o.”, 
where @, is given by (4), and @, by (10), with & omitted. 
If the effect of (I’) is small we should get from (4) and (13) 
1 GOR Ma*l?(3a+46) _ 3 Va(a?—20°) (17) 
12E1& 4 E103 (3a+46)"* 


wo? 2377 El” 
In this case I have not worked out an independent dynamical 
method taking both shaft and load into account; but the 
difficulties would be less than in case (3). 
Case 5 (Dunkerley’s Cases V. and XIII). 
§ 28. Shaft supported at ends A and B, and at inter- 
mediate point O 
(WA—a« .OB=—6,' a+b=1). 
A O Bo 
«8 
(a) Unloaded shaft: Euler-Bernoulli solution. 
As pointed out by Dunkerley, the mathematical conditions 
are all satisfied if the two relations (ef. § 12) 
pa = it, 
basa ¢ Mere ake ee 1.) 
where z and 7 are integers, can exist simultaneously. 
To have a real application to the practical problem, 7 and j 
must be small integers, so that (1) is of very limited scope. 
If, however, the spans are equal, or if the longer, say a, is a 
multiple of 6, we have obviously 
Pe oe. A. 
Answering to this, we obtain for the critical velocity 
wo = EI/opl* = 97-41 Hl /opi*. . . . (3) 
This is the same result (¢/. (a) Case 2) as for a shaft of length 
5 supported at both ends. 
Excluding the above special cases, the general equation 
obtained by Dunkerley (/.c¢. p. 296) is 
coth wa—cot wat coth ub—cot wb=0. . . . (4) 
