280 
MR. S. DUNKERLPJY OY THE WHIRLING 
Art. Page. 
21-22 Second metLocI of solution. Modification of general theory. Investigation con¬ 
fined to effect of one pulley only, the effect of shaft being neglected . , . 302 
23-25 Case IX.—Overhanging shaft fixed in direction at one end and loaded -Ruth a 
pulley at the free end. Superior and inferior limits. Natural period of 
vibration. Result,s.303 
26-30 Case X.—Shaft resting on a bearing at each end and loaded with a pulley. 
Superior and inferior limits. Natural period of vibration. Theoretical 
results. Experimental results.306 
31-34 Case XI.—Shaft, resting on two bearings and overhanging on one side, loaded 
with a pulley at end of overhanging portion. Superior and inferior limits. 
Natural period of vibration. Theoretical results. Experimental results . . 313 
35-38 Case XII.—Shaft, fixed in direction at one end and resting freely on a bearing at 
the other, loaded with a pulley. Supeiior and inferior limits. Natural 
period of vibration. Theoretical results. Experimental results. Comparison 
with Case X.320 
39-46 Case XIII.—Shaft resting on three bearings and loaded with a pulley on one of 
the spans. Case of equal spans. Superior and inferior limits. Natural 
period of Adbration. Theoretical results. Comparison with Case X. Experi¬ 
mental results. Remarks on formula N;^N 2 /■v/(Np + No“).325 
47-49 Case XIV.—Shaft fixed in direction at each end and loaded with a pulley. 
Superior and inferior limits. Natural period of vibration. Theoretical 
results. Comparison with Cases XII. and X.336 
50-54 Case XV.—Shaft supported on four bearings and loaded Avith a pulley on one of 
the outer sjoans. Case of equal spans. Superior and inferior limits. Natural 
period of Aubration. Theoretical results. Comparison with Case XIII. 
Stiffening effect of a fourth bearing.340 
55-58 Case XVI.—Shaft supported on four bearings and loaded with a pulley on the 
middle span. Case of equal span. Superior and inferior limits. Natural 
period of vibration. Theoretical results. Comparison Avith Cases X., XIV., 
XIII., and XV. Stiffening effect of a fourth bearing-.348 
59 Case of two or more pulleys .355 
60-62 Case XVII.—Shaft resting on a bearing at each end and loaded Avith tAvo 
pulleys, the effect of the shaft being neglected. Equations too complicated 
for actual use. AlternatiA'e method. Formula for calculating the resulting 
u'hirling speed due to several causes, in terms of the whirling speeds due to the 
several causes considered independently of on? another .355 
63 Concluding remarks.359 
CHAPTER I.—INTRODUCTION AND DESCRIPTION OF EXPERIJtlENTAL APPARATUS. 
Introduction. 
1. It is well known that every s\\txitJiou'ev€r nearly balanced, when driven at a par¬ 
ticular speed, bends, and, unless the amount of dedection be limited, might even break, 
although at higher speeds the shaft again runs true. The particular or “ critical ” 
speed depends on the manner in Avhich the shaft is supported, its size and modulus 
of elasticity, and the size, weight, and position of any pulleys it carries. 
