118 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



which is the general equation of pendular motion, the frequency being very nearly 

 that given in Chapter I. 



In order to get a still better approximation we can insert for y^,m (4), the value 

 of y just found in equation (6), and using the value just obtained, (7), for yi, 

 whence — 



_^ = ip^- yl' JY. 08164/^ - . II2698 IH + .0416 Z' 



-.01— + . 000198-^) (x-z) dz 



V If 



= i.|l^ 3,; (. 04097 /* x' -. 018783 /^ x^* 4- . 00138 x" 

 E J I \ 



x'' x^° \ 



-.000264— +.000002 -rj I 



I I / 



or, integrating, 



-y = ^^j^3'i (-663084 /«-.9i27i8rx + . 341435^'^" 



-.093915 /'^x^ + .oo248x - .000367 — + .000002 -jj). 



Thus the vibration of the free end (where ;r^o) becomes — 



- 3,j = .o8o9227— f-y, 

 rL J 



whence the frequency — 



^_ 3-5153^ /"^ 



which is still nearer its true value, although the difference is not great. 



3. Free-Free Bar. — If both ends are free, and there are no external forces act- 

 ing on the bar, the end conditions at both ends are the same — 



(Py d^y 



dx^ dx^ 



In other words, the extreme deflection is that value, yi, in function of which we will 

 express the elastic line; furthermore, at the ends there is no bending moment; and 

 no shear. Assuming the deflection to be given in function of the ordinate by some 

 such equation as — 



y =^ A + Bx -\- Cx' -\- Dx' + Ex' + Fx' -f Gx', 



we can readily see that at one end, where x = 0, we find A =yi ; also C == D = o ; 

 the conditions at the other end {x = /) give (forming the second and third deriva- 

 tives and applying ordinary rules of determinants) — 



B= -^GP E = ^GP F=- 2,Gl. 

 2 2 



