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IX. The Transverse Vibrations of a Rod of Varying Cross- 

 Section. By P. F. Ward, M.Sc. (Manch.)* 



1. ri^HE following paper contains an\extension o£ some 

 JL results given by Kirchhoff in his memoir on the 

 vibrations of a bar of variable section f. It is assumed that 

 the cross-sections are small, their centres of gravity in a 

 straight line, and their principal axes in the same directions. 

 Such a rod can execute small vibrations parallel to either 

 of these two directions. The rod is further supposed to be 

 of rectangular section. 



After the differential equation has been formed, it is 

 simplified by making a special assumption as to the law of 

 variation of the cross-section. It is then solved in terms of 

 Bessel's functions, by an extension of the artifice used by 

 Kirchhoff in the case where the rod has the form of a very 

 acute-angled wedge or cone. The thin end of the rod is sup- 

 posed to be free, and the solution is investigated for the cases 

 where the other end is clamped, free, or supported, respectively. 

 The lower modes of vibration are then investigated in the case 

 of the wedge and cone. As Kirchhoff found, the amplitude 

 of vibration can become relatively very large near the thin 

 end of the rod. It is also shown that there is an approximate 

 relation between the positions of the nodes, points of maximum 

 excursion, and points of inflexion, and the roots of certain 

 Bessel's functions, the approximation being closer the higher 

 the mode, and the nearer the points in question to the thin 

 ond of the rod. 



A discussion is appended of some cases of wave-motion in 

 a canal of variable cross-section, which are easily solved by 

 means of Bessel's functions. 



2. Take the line of centres of the cross-sections in the 

 equilibrium position as the s-axis of a rectangular system, 

 the axis of x being in the direction of vibration. Then, if 

 E denotes Young's modulus, a> the area of the cross-section, 

 jc the radius of gyration of a section about an axis through 

 its centre parallel to y, f the transverse displacement of the 

 centre of a section, and /j the volume-density, the equation 

 of motion is 



" <a 9?=-^( W P)- ■ • / « 



* Communicated by Prof. H. Lamb, F.R.S. 

 t Gesammelte Abhandlungen, p. 839. 



