﻿[ 233 ] 



XXXI. On the Lateral Vibration of Bars subjected to Forces 

 in the Direction of their Awes. By Johx Morrow, 

 M.Sc.(VicL), I).ErHj.(L'pool), Lecturer in Engineering, 



University College, Bristol *. 



Contents. 



Section I. Statement of the General Problem. 



„ II. Supported Massive Bar, Axial Tension. 



„ III. Clamped-CJamped Massive Bar, Axial Tension. 



„ IV. Clamped-Supported „ „ „ „ 



„ V. Supported Loaded Bar of Negligible Mass, Axial Thrust. 



„ VI. Clamped „ „ „ „ „ „ 



„ VII. Deduction of Solutions to some Static Problems. 



Section I. Statement of the Genera/ Problem. 

 § 1. The general differential equation for a rod of uniform 

 or varying sectional area subjected to an axial tensile force 

 may be written, neglecting the rotatory inertia, 



a?l BI aJ)- p si + '-3F =0 -- • • (1) 



In which 



y= lateral displacement at a distance x from the origin at 

 time t, the axis of x being parallel to that of the bar, 

 and vibration occurring in the xy plane : 



p = density of the material ; 



w = sectional area at x : 



P = total axial pull; 



I = Moment of Inertia of section at x about a line through 

 its centre of gravity at right-angles to the plane of 

 vibration ; 



P 



E= (-Young's Modulus (sensibly equal to Young's 



Modulus for the material, which is assumed homo- 

 geneous and isotropic). 



§ 2. If the bar carries loads concentrated at different points 

 in its length, equation (1) holds between every pair of singular 

 points, that is between points of support and load or between 

 any pair of concentrated loads. 



If undashed symbols refer to values of y on the left, and 

 dashed symbols to those on the right of any singular point, 

 we have, in addition to the end conditions, that at all singular 

 points 



di, dy' 



y=y' and 



ix~ dx 



These serve to determine the constants in the solution of (1). 

 * Communicated by the Physical Society : read April 27. 1906. 



