[ 81 ] 



IV. The Transverse Vibrations of Beams and tlie Whirling 

 of Shafts supported at Intermediate Points. By E. It. 

 . Darnley*. 



§ 1. npHIS paper investigates the equations giving the 

 JL periods of the transverse vibrations o£ uniform 

 beams and the whirling speeds of uniform shafts, when the 

 beams or shafts are simply supported at any number of 

 intermediate points (§§4-6). The remainder of the paper 

 is confined to the case where the ends also are simply sup- 

 ported. New functions denoted by <f> and y\r are tabulated 

 and graphed. The use of these tables and graphs considerably 

 extends the class of cases in which numerical solutions can 

 be obtained, and suggests that some of Dunkerley's results 

 need revision (§§ 7-10). A general theorem is given 

 relating to symmetrical arrangements of the supports (§ 11). 

 This theorem is similar to one relating to the critical loading 

 of a strut, recently published by (<owley and Levy "j\ The 

 question of the distribution of the supports to give the 

 greatest value of the slowest period or whirling speed is 

 discussed ; and it is shown that this period or speed is a 

 maximum when the supports are equidistant (§ 13). 



The general theory of the vibrations of beams of one 

 section or bay has been treated by Kayleigh in chapter VIII. 

 of his book on Sound, and by Love in chapters XII. and XX. 

 of his book on Elasticity. The former gives a detailed 

 account of the vibrations of a beam in the six possible cases 

 of terminal conditions, arising according as either or both 

 of the ends are supported, clamped, or free, with Yerj exact 

 numerical solutions for the periods, and a study of the shape 

 of the vibrating beam and of the position of the nodes and 

 loops in certain of the graver modes of vibration. The latter 

 gives an account of the flexural vibrations of a circular 

 cylinder from the point of view of the mathematical theory 

 of elasticity, lending to a complicated frequency equation 

 containing BessePs functions, which reduces, when the radius 

 of the cylinder is supposed small, to the equation derived 

 from KirchhofPs theory of thin rods or beams, which is used 

 by Rayleigh and in this paper. 



The theory of the transverse vibrations of thin beams is 

 analytically identical with that of the whirling o^i shafts. 

 regarding which reference may in particular be made to the 

 papers by Dunkerley and Chree, quoted in the footnote. 



* Communicated by Prof. A. N. Whitehead, F.R.S. 

 f Troc. Hoy. Soc. vol. xciv. p. 405 (1918). 



Phil. Mag. S. 6. Vol. 41. No. 211. Jan. L921. (i 



