116 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



If, for instance, y^ be the displacement at a distance z from the point chosen 

 as origin, so that — ^ is the acceleration, and if y^ be the displacement of any 



chosen point (say that at which the displacement is a maximum) then the above 

 statement may be expressed by the equation : — 



^/^.= ^/j. or simply, yl = | 3'. (x) 



This, of course, will be instantly recognized as the general equation of vibratory 



motion, d^y/df = — o'v, of which the frequency is = — or the period is = — . On 



2n a 



the other hand, we know the expression M = EJ/ p connecting the bending moment, 

 the moment of inertia of the cross-section, and the radius of curvature at the sec- 

 tion in question. This is often given in a differential form (assuming that the radius 



1 ld'y\ 



is very nearly = '^j^r^ ) 



In a freely vibrating beam or bar the moment M is the restoring agency ; in other 

 words, a couple due to reversed effective forces of extension and compression, act- 

 ing in any cross-section according to the well-known straight line theory; E, of 

 course, is the modulus of elasticity, and / is the moment of inertia, about the neu- 

 tral axis of the cross-section. Now, since the moment M is a function of the mass 

 per unit length, of the end forces, and of the accelerations, we can express it in terms 



of ^ y^ and the density and cross-sectional area of the bar. And assuming for 



y^ a hypothetical type of vibration, satisfying the end conditions, we can insert the 

 value of M thus obtained in the equation (2). Solving for y we have a second ap- 

 proximation to the center line of the bar, and the period can be calculated as 



= ^n^-^ (3) 



T 



Ji 



The value of y just found can now be substituted into (2) and a closer approx- 

 imation secured by solving the results, as will presently be illustrated. Professor 

 Morrow's paper should be carefully studied in detail ; but for our immediate purpose 

 we shall take only the following examples :' — 



2. Clamped-Free Bar. — Let the origin be at the free end of the bar, whose 

 length is /, the area of cross-section being a and the density of the metal being y 

 By D'Alembert's principle the acting forces are in equilibrium with the inertia 

 forces. Now the moment of the inertia forces about some section X (distant x from 

 the origin) is the sum of infinitely small products: — mass times acceleration times 

 leverage. 



jyayl {x-z) dz\ 



