114 Mr. Morrow on Lateral Vibration of Bars of 



from instantaneous photographs of the vibrating bar, has 

 concluded that a better approximation to the centre-line of 

 the bar is obtained if the lateral force is assumed to act at a 

 point one-fifth of the length from the free end. 



Kayleigh's method is, of course, applicable to rods under 

 other conditions of support. It rests on the fact"* that a 

 close approach to the true period can be obtained by assuming 

 a type of vibration which is admissible as an initial con- 

 figuration, as a considerable departure from the true type 

 leads to only a small error in the estimate of the frequency. 



Garrett's method and results have been further investigated 

 and compared with Lord Kayleigh's by Dr. Chree f , and it 

 is shown that, whilst Kayleigh's depends on a recognized 

 dynamical principle, Garrett's apparently has no such basis. 



These simple solutions may be of considerable importance 

 in acoustics and mechanics. They have been employed in 

 problems of the u whirling " and vibration of rotating shafts 

 by Dunkerley J and later by Chree § . 



It is the object of the present paper to show that by 

 assuming an equation which completely satisfies the end 

 conditions a better approximation is obtained, and that, by a 

 process of continuous approximation, the vibration curve and 

 the period of the fundamental are given to any required 

 degree of accuracy. 



The treatment is capable of very general application, and 

 will be chiefly useful in problems of which the solution has 

 not previously been obtained. The first three cases dealt 

 with will serve as illustrations under various types of terminal 

 conditions. It is a noteworthy fact that, whilst Kayleigh's 

 approximations are necessarily overestimates, the method 

 described below gives initially too small a value for the 

 frequency. 



When the density and flexural rigidity of a bar are 

 variable from point to point in its length, its treatment by 

 the ordinary method of analysis presents difficulties which 

 have not yet been completely overcome. Many of the 

 solutions are obtained easily by the method here given, and 

 several special cases are considered. 



2. Approximate Method of Solution. 

 If a bar is vibrating so that every point in it has the same 

 period, the ratio of the acceleration to the displacement is 

 constant for all points. 



* Kayleigh's ' Sound,' vol. i. §§ 88, 89, 182. 

 t Phil. Mag. Jan. 1905, p. 134. 

 X Phil. Trans. A, 1894, p. 279. 

 § Phil. Mag. May 1904, p. 504. 



