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XIII. On the Lateral Vibration of Bars of Uniform and 

 Varying Sectional Area. "By Johk Morrow, M.Sc. ( Vict.) ; 

 M. Eng. (Liverpool) ; Lecturer in Engineering, University 

 College, Bristol*. 



Contents. 



1. Introduction. 



2. Approximate Method of Solution. 



3. Clamped-Free Bar. 



4. Bar " Supported " at Both Ends. 



5. Free-Free Bar. 



6. Clamped-Free Bar of Varying Breadth (b = A.r). 



7. „ „ „ (b = Ax 2 ). 



8. „ „ Depth (d=Ax). 



9. Kirchhoff's Investigation. 



I. Introduction. 



IN the theory of vibrating rods expressions for the 

 frequency and the form of the displaced bar are ob- 

 tained from the equation 



2?.= ^' 



in which y is the displacement at a distance x from the 

 origin. 



The solution is 



y = A sin fix -f B cos fix + C sinh fix + D cosh fix, 



A, B, C, D being arbitrary constants to be determined by the 

 end conditions. 



In the case of a uniform bar supported at each end the 

 numerical solution presents no difficulty. In other ele- 

 mentary cases the results are deduced from the values of 

 the roots of equations which would be troublesome if not 

 previously tabulated. 



Lord Bayleigh f has shown that the natural period of a 

 bar may be obtained approximately by simpler means. For 

 example, he supposes that the curve assumed by a clamped- 

 free bar whilst vibrating is the same as that which it would 

 take up if displaced from its position of equilibrium by a 

 lateral force acting on the bar at some point in its length. 

 He shows that the period is given with considerable accuracy 

 if the point of application of the force is one quarter of the 

 length from the free end. 



More recently, Garrett { has investigated this subject and, 



* Communicated by the Physical Society : read March 10, 190'5. 

 t See Kayleigh's ' Sound/ vol. i. pp. 233-235. 

 X Phil. Mag. Nov. 1904. 



Phil. Maq. S. 6. Vol. 10. No. 55. July 1905- I 



