AND VIBRATION OF SHAFTS. 
281 
The theory for the case of an unloaded shaft first received attention at the liands 
of Professor PiANKINE/“ who obtained numerical formulae for the cases of an unloaded 
shaft resting freely on a bearing at each end, aud for an overhanging shaft working 
in a shoulder at one end. 
Professor Geeenhill has also obtained formulae for the cases of an unloaded shaft 
resting on bearings at each end, and fixed in direction at each end.t 
The theory has been further extended to the case of a shaft loaded with pulleys, 
by Professor Keynolds ; and the object of this investigation is to apply that theory 
and so obtain formulae, and by experiment to verify them, giving the critical speed 
in terms of the diameter of the shaft, weights of pulleys, &c., in particular cases 
applicable to the different conditions under v/hich a shaft works. 
In many cases, as might naturally be expected, the “ period of whirl ” of the shaft 
is merely its natural period of lateral vibration when in a state of rest. The two 
periods are coincident in the case of an unloaded shaft (however supported), and for 
a loaded shaft on which the pulleys are placed in such positions that they rotate-.- 
when the shaft is whirling—in planes perpendicular to the original alignment of the 
shaft. With pulleys placed in any other positions, when the shaft is whirling, there 
is a righting moment, tending to straighten the shaft, which does not exist when it 
merely vibrates under the dead weight of the pulleys. 
Hence, in an unloaded shaft, the period of whirl coincides with the natural period 
of lateral vibration ; but, generally, in a loaded shaft, the period of whirl is less than 
the natural period of vibration, to an extent depending on the size and positions of 
the pulleys. 
If, therefore, the period of disturbance (that is, the period of one revolution) be 
decreased, the shaft runs true until that period approximates to the natural period of 
vibration of the shaft (assumed at rest) under the given conditions. If the shaft 
now receive any displacement, however slight, a violent agitation is set up, which 
will be most marked when the period of disturbance and the whirling period coincide. 
As the period of disturbance is further decreased, the agitation becomes less, and, at 
a period of disturbance slightly less than the wdiirling period of the shaft, the shaft 
will again run true. 
As in the vibration of rods, so in the whirling of shafts, there are a series of 
periods at which the shaft whirls. 
Experimental Apparatus. 
2. The experiments were made in the Whitwmrtli Engineering Laboratory, 
Owens College, where the essential facilities for obtaining uniform rotation at any 
MDCCCXCIV.—A. 
* R.^nkine’s ‘Machinery and Millwork,’ p. 549. 
t ‘ Proc. o£ Inst. Mech. Engineers,’ April, 1883. 
2 O 
